I must admit that in my basic GA training, we did not do spins. I had to go and do an emergency recoveries course with Matt Handley in the S2B. We trained in Tomahawks, but they didn't want to do spins due to the strain on the airframe (even though it was certified). At least on the Tommy, a stall was pretty much guaranteed to drop a wing each and every time, so it was good training anyway.
I was taught that the spin is not stable in most aircraft for the close to the first turn anyway, if not more. So the first rotation was considered an incipent spin, and referred to as a stable spin once it was "stable". That was with a normal stall entry into a spin, the same kind of unrealistic "slowly decelerate, maintaining level flight until stick is full back" stall, followed by rudder. A clean and stable spin entry in aerobatics was done quite differently. And I would suggest that most of the stall/spin accidents are also entered quite differently. I've commented on a few threads about what new privileges RAA should strive for, but if there was one I would back for RAA and GA, it is to bring back spin recovery for everyone. Entered from the sorts of attitudes likely to be a problem, like climbing turns when the engine quits.