When people talk about a classic wrist shot, they're referring to how wrist shots used to be taken, with wood sticks, where you'd really draw the puck back and then sweep it forward, rolling the wrists at the end, but with the puck constantly in contact with the blade.
Classic snap shots were simply slap shots with very small windups, sometimes without the blade ever leaving the ice, but essentially any sort of shot where you'd flex the stick behind the puck, before it come into contact with the puck, but not slap shots, which have bigger wind ups.
Today, most people take shots that are essentially a hybrid of classic wrist shots and classic snap shots. These "modern wrist shots," or wrist/snap hybrids, became essentially the only type of wrist or snap shot used once composite sticks got popular, but were around even before that. It's where you pull the puck a little in and forward, let it get slightly ahead of your stick, then really focus on snapping the stick and shooting using the flex of the shaft for power, all in one smooth motion. Not quite a classic wrist shot or a classic snap shot, but a sort of combo of both, sometimes people just call these simply wrist shots OR snap shots (confusing, they used to be very different but have now come to mean the same thing), sometimes "modern wrist shots" or "modern snap shots", but it's basically how all good players shoot nowadays for all non-slap shots. You can take them off the inside or the outside foot. The mini-wind-up slap shots, that used to be called snap shots, are now just considered slap shots.
Modern wrsit shot/modern snap shot (same thing basically):