It's certainly true that different types recover differently. I've done dozens, maybe over a hundred, spins in the Pitts S2C. It is a pussy cat to recover. Opposite rudder, relax the stick, and it's out in a quarter turn even from a fully developed (>3 turn) spin. But recently I flew the Extra 300 and there you have to do it by the book. Opposite rudder, wait, wait, then stick full forward. It WILL recover if you do it the Pitts way, but only after half a turn or more. It's said that the "let go" technique doesn't work in the Extra, though I haven't tried.
Flat spins (which you should never get into by accident, if you're in CG limits) are a different matter. It's kind of fun to watch the airspeed drop to zero. In the Pitts, a "proper" recovery takes about three turns while you wait for the nose to drop, the airspeed to build and the rudder to get some authority. BUT... you can cheat. If you leave the power in (which you need to get the spin to flatten in the first place) then the propwash gives you enough rudder authority to recover almost instantly.
That works fine... until it doesn't. I got so used to the technique that I routinely recovered spins without pulling power. Then I got into a "knife edge spin" which feels like a spin but isn't really. Nothing I did helped. My instructor pulled the power and we dropped out of it in an instant (it's maintained gyroscopically). Later I got into one again (I have no idea how to enter one deliberately, and neither does my seriously experienced instructor). Following my new training, I instantly pulled the power and recovered. "Oh," he said, "I was hoping you'd hold it for a while and see how it developed".