As far as I know, a draw cut is defined as slicing the opponent by moving the blade against his body, rather than swinging at him. That's why I said it's a move of opportunity/desperation - if your sword is in contact with the opponent, but trapped so that you can't pull it back, this is better than nothing and might free it. This is clearly not what he was talking about.He is clearly talking about drawing the sword back as you cut, or, in other words, a "draw cut."
It's swung no differently than a longsword except you can't do reverse edge cuts. Literally, pick up a katana and do all of the longsword cuts you would normally do, but don't do any reverse edge cuts because it obviously doesn't have a reverse edge. The katana will perform pretty much exactly the same as the longsword and pretty much exactly the same as if you were doing your special Edo period swordsman cuts. Obviously edge alignment is important, I never suggested it wasn't, and at this point you are deliberately not engaging with the argument whatsoever and just inventing things I never said, and I'm done arguing with you.
I'm not experienced with a katana, so I couldn't realy say, but you seem awfully confident for somone who disagrees with the bulk of literature on the topic. Stubborn and easily offended when your preconceptions are challenged too. You seem to have beef with the Edo period, but I'm starting to think you'd fit right in.