I am by no means a falafel expert, but I think my falafel come out pretty damn good.
Falafel fall under that regional category where everyone has their own idea of the correct recipe. I've seen this happen with other regional dishes, where it can vary wildly from town to town, or country to country.
The answer is, they're all "correct" and "authentic". Woe to anyone who accuses a grandmother of making a dish "wrong" just because it's not the exact same recipe as their own.
I make a chickpea falafel, but I have seen variants that include broad beans. I'm comfortable enough with the food that I believe I'll try other legumes in the future. I used Saveur's recipe for a base, and added a jalapeno and cilantro to the mix. I've never used canned beans, so I don't know how different the texture would be compared to soaked beans.
If you can't get the dough to hold together, try smaller balls, or add a little bit of flour, a couple tablespoons at a time, until they do. Coconut flour is a good binder if you're gluten free.
I like them in a pita, or as a fancy salad topping with the tahini sauce as a dressing.