It can depend on what type of game you are playing, why you use 'pocket speed' and what part of the world or even country you are from as to how you describe what pocket speed means.
To me I use it mostly in 8 ball as a tactic to purposely miss a pocket but leave it over blocking my opponents balls. So if I think to my self I need to hit this pocket speed into the side rail to leave it over the pocket I hit it at a speed that if I were to take the pot on, and make it, the ball wouldn't hit the back of the pocket but simply drop over the shelf.
In 9 & 10 ball its more of a result of getting out of position and being too thin on a ball. If I need to limit the CB travel after a shot ill hit a shot 'pocket speed'. To me this is when the ball doesn't hit the back of the pocket, but is in no danger of falling short.
Others will use it for other reasons mainly, so their opinions on how to best describe it will vary. As long as you know what it means to you, and why you use it.
As for pocket speed meaning a larger pocket, yes and no. Depends hugely on the table you play on. Generally as a rule of thumb yes, potting an object ball at slower speeds means it will have less chance of rattling. We have a club around here with a very tight diamond table so people generally play shots slower. The thing with this table is its on a floor that's not even so it rolls off a lot so that's just one example how it can screw you over playing pocket speed. Another is an English pool table I used to own. It had pockets that were set further back than any I've seen before so even when a frozen ball on the rail was hit with just enough speed to drop it would hang up with the rounded jaws. However if I hammered the same shot the ball would drop, it kind of rattled in the jaws, hopped in the air and into the pocket. So you could say pocket speed on that table for a frozen rail shot was hard.
