The correct answer, according to many, has more to do with the way it’s paddled and little to do with the boat.
Almost by definition, if you are paddling with a canoe paddle, it’s a canoe.
If you are paddling with a kayak paddle, it’s a kayak.
Hence, a decked canoe, though very much like a kayak, is a canoe, if it’s paddled with a single blade paddle.
When I ordered my Wenonah Voyageur with lower seats, for paddling with a kayak paddle, it was considered pretty unusual. You would have had a fairly hard time finding photos of people paddling open top canoes with a double bladed paddle. They were out there, but it wasn’t widely accepted.
Sensibly, that has changed and most people refer to open top boats as canoes, even if they are paddled with a double blade paddle, and many canoe companies have photos of their boats being paddled with canoe paddles and even sell versions designed specifically for that.
I still have a hard time calling a decked canoe a canoe. If I paddle it with a kayak paddle, does it become a kayak?
It gets more confusing when using oars… 
The paddle doesn’t change the boat.
A canoe is a canoe if you use a canoe paddle, kayak paddle, motor, or your hand
Decked canoes are still canoes, or they would be called kayaks.
The line that might differentiate a canoe from a kayak can be a fine line within a big grey area. 
I paddled a Clipper Sea-1 decked canoe for years and people would often ask me if it was a kayak or a canoe. The partial deck and raised canoe seat was the confusing part because with a rudder and the extended hull shape it sure looked like a sea kayak. It could be paddled with either a single-blade or double-blade but I found that I much preferred paddling with a canoe paddle. A fine line here.
The traditional pack canoe is a canoe in hull shape and lack of deck but with a low (or no) seat it’s not meant to be paddled with a single blade paddle. So is this an open kayak? Another fine line.
A C1 slalom canoe is certainly a kayak in hull shape and covered cockpit but it’s a canoe by virtue of the pedestal seat and single-blade paddle. A really fine line here.
And then there are rafts with elongated hulls that can be paddled with either a double blade or single blade. What the heck are these?
Maybe the distinction is defined by the seating configuration: Raised seat = canoe and Lowered (or no) seat = kayak. But raised by how much?
Much ado about not much. If you need pictures of canoes being paddled double bladed, here. Gallery - Placid Boatworks
You get the boat that accomplishes what you need to do. Apparently in your case that meant you had to look at boats of which you were not aware. You can call it whatever you wish, as long as you are describing it clearly in conversations the label is just a word.
I have a Placidboat Rapidfire CANOE. I know they are marketed to paddlers who most likely will use a double blade kayak paddle and don’t care to bother with the learning curve of obtaining single blade skill. I originally had the triple stack of floor mounted seats, but wen I saw Joe’s (the PB owner) own canoe with an elevated rail mounted seat, I asked to have one installed in my canoe so I could get off the floor and be better placed to use a single blade CANOE paddle. He initially discouraged the idea, since his seat was designed for racing and he did not then know me well. Well, I am a racer too. I paddle my RF with just one exception 100% entirely with single blade paddles and love it. I even use an additional seat pad or two to elevate me a little more.
The only exception to paddle use is during official races (like the ADK 90 miler), wherein the only competitive class that canoe fits in (solo recreational) the rules require use of a double blade paddle. I do it when I enter races solo, but I hate it. I miss the fine control maneuverability stroke variety I get with a single blade, not possible with a clunky kayak paddle and when doing the high angle power strokes, necessary to compete while racing, causes drips to fall inside the canoe with every stroke. I once paddled along beside another RF racer who regularly had to stop in the shallows to dump gallons of water out, while I had a fabric spray cover on mine which prevented that problem. Othewise I am a dedicated single blade paddler in all canoes.
The Placid boats do an always have been able to be set up both ways. Your practice with them is one of those options.
The OPer indicated there were no photos of essentially undecked boats being paddled with double blades. I found some.
My wife paddles her canoe with a double blade. Very common with a pack-style canoe.
A canoe is what I have and a kayak is what I don’t want.
If it sits low in the water and is designed to have water come over the bow, to penetrate waves, to me it’s a kayak.
Look to the world of bikes they have a dozen different names. I like a hybrid best.
They say Eskimos now called Indigenous Circumpolar Peoples have 50 words for snow.
It becomes confusing to us newcomers to paddle boats as both canoes and kayaks have several sub groups they seem to overlap also. I clearly bought a canoe OT Guide 147. I stripped it out making it a solo and building a center sit back seat that sits a tad lower than original and I use a longer kayak paddle. All the outside ID is removed and people have asked what it is. It looks a bit like a pack canoe but wider and heavier.
I think from now on I will call it a hybrid canoe, just like I call my mountain bike I have modified for street touring a hybrid bike. 
As things become more specialized…and people are doing more, there are more hybrids out there. In the 80s I had a Trek touring bike that I used for distance, commuting, trails and urban “trials” type of play.
I called it “a bike” 
All around good canoe.
pic didn’t work due to being sold and old.
There is no clear distinction.
Generally speaking, canoes lack full decks and are either open or have partial decks. But then sit on top kayaks lack decks altogether and C-1 whitewater canoes have full decks.
Pack canoes are pretty close to deckless kayaks.
Most canoes are paddled either kneeling or sitting at least a half dozen or more inches above the hull bottom. Except for pack canoes which are often paddled sitting on the floor, or nearly so.
Kayaks are most often paddled with double bladed paddles but can be paddled with single bladed paddles. Canoes are most often paddled with single bladed paddles but can be paddled with double bladed paddles.
Gets even more confusing when you cross the Atlantic, as the Brits call both over there a canoe. What we call a kayak they call a kayak canoe (often shortened to just canoe). What we call canoe, I think they call American Canoe (also often shortened to just canoe).
Well, it comes awfully close to being a distinction without a difference, doesn’t it? We’re talking about boats, in either case - not the way they’re paddled. Personally, I think its the seat height that differentiates, but its sure not worth arguing over. IMHO, raised its a canoe no matter how the paddler chooses to propel it. There’s always been a lot of grey area.
Heck, in Europe they call 'em all canoes and call it a “Canadian canoe” if its open. Its more a distinction of language than boat. If that language usage is accepted, then a kayak is a specific type of canoe.
Way back when the old Rushtons (open and double blade used - Sairy Gamp) as built for “Nessmuk” walked that line. They would now be called pac canoes. But the Rushton Wee Lassie had a deck and over time boats called a “Wee Lassie” have gotten increasingly more deck and higher seat placement. They are now commonly paddled with a single blade. That was the boat that did the most to popularize canoeing in Europe, if my understanding of this is correct.
When I was young Folbots were pretty common, sold from magazine ads all over the place. They had decks - and folks usually, but not always, paddled them with double blades. Probably more kayak than canoe but with a larger cockpit than is normally associated with kayaks. And what of the Krugers? They have about as much deck as the Folbot but are usually considered canoes - again, its the seat height that seems to make the difference if a distinction must be made.
I’m not a racer and don’t know anything about the racing class systems that are, or have historically been, used but aren’t there different aspect ratios or max/min specifications for the classes that would separate canoes from kayaks in racing events on that basis?
And, thinking of Rushtons and history, what differentiates a sailing canoe from a sail boat? Maybe all sailing canoes are sailboats but not all sailboats are sailing canoes.
Venn diagrams for canoes, kayaks, pac boats, sailing canoes, and sail boats anyone?
"You would have had a fairly hard time finding photos of people paddling open top canoes with a double bladed paddle. "
Not at all. A quick search string turns up hundreds of such photos. I’ve paddled tandem and solo canoes with both single and double blades for decades.
Of course the distinctions between kayak and canoe in popular common language are pretty fuzzy by now. Some of us are historical purists and consider a true kayak to have the characteristics of a traditional skin boat as developed by native peoples of the northern regions, in other words, a long, fairly narrow and low profile decked boat with a fairly compact cockpit that one sits inside, near or below the water line. It’s a kayak (qajaq) no matter if it’s propelled with a single or double blade or even just a pole. I’ve never considered sit on tops as really kayaks – most of them have more in common with rafts or fishing skiffs in being wide and open. Some of them approach the narrow fast design of surf skis, which is a whole other category. And the distinction among sit on tops, “rec kayaks”, SUPs and even surfboards becomes more vague all the time. (I’ve been known to refer to cheap rec boats as “quasiyaks”…)
I think the most distinct characteristic of canoes is that they tend to have more lateral symmetry and equal height stems, many can even be paddled facing from either direction, whereas kayaks in general have a definite fixed bow and stern. A canoe will typically have thwarts and a yoke too.
But, as The Bard has young Juliet wisely observe “That which we call a rose would by any other name smell as sweet.”
Good discussion. I agree with one notable exception. Not all canoes are symmetrical so to be paddled in either direction. Fishform canoes are wider in the Bow and Swedeform are wider in the Stern. So many exceptions!!!
Much ado about nothing, until…One sits on the ACA SEIC and participates in revising the various instructional curricula. This committee was composed of nationally known paddlers in all fields and this very question came up. How many do we need? Particularly vexing is an Olympic class known as C-1. This is a WW kayak hull with skirt in which the paddler kneels with a single blade paddle. As the discussion proceeded , it became less clear and more confusing. This seemed to be more of a rhetorical question than anything. We finally passed a Kayak and Canoe curricula allowing individual instructors to choose which hulls to allow in their classes.
And decked canoes or partially decked canoes often lack thwarts.
One distinction not mentioned and it is as clear as mud as all the other things are.
You wear a kayak. It should fit as finely tailored pants
You don’t wear a canoe. You have sone slop room
Here we go into the murk of canoes with thigh straps
There was some excitement paddling into the sea yesterday. There was a gale warning, but I don’t think the winds were quite up to gale force yesterday morning. The wind was strong. It was both onshore and alongshore, but it was definitely onshore and not offshore. And it had just picked up the evening before, so it hadn’t been blowing sustained onshore for very long. These were short-period waves to be sure. Every time I paddled out, my face was met with everything from splashes to heavy slaps of water, with broken waves washing significant volumes of water over my deck. Several times riding in, I had water washing over the decks. And once off the beach, the waves remained steep, with the occasional longer whitecaps to go along with it. The combination of wind, short period waves, and whitecaps demand a little attention even in deep open water, and there’s no eddy to park in while you would attempt to bail. So while I’m not saying an adventurous expert in a canoe couldn’t do it, I don’t know that I could be convinced that their canoe is the best design for this type of thing. While not being so purist as to feel a proper kayak must be skin-on-frame, I feel like a proper kayak should allow the decks to be awash without subsequent bailing, and you should be able to navigate your boat from place to place effectively in these conditions without your boat becoming swamped. It should be rollable without becoming swamped. I figure it should be an open hull where you sit inside of it, and sit-on-tops fall into their own category, such as wave ski and surf ski.
My thought is that a proper canoe allows for a raised center of gravity, and all of the stuff, pros and cons, or all pros, or all cons (depending upon whom you ask), that go along with that. I think that a canoe that is covered is typically referred to as covered, as opposed to a canoe’s typical state of being, and as such, I see a canoe as a non-decked, open hull paddlecraft designed to allow a variety of paddling positions, and of course, you can add covering per your needs. So I guess I think canoeing is a lot about the paddling position(s).