in this context, they're just terms that manufacturers can use. they have no real meaning in of themself. they only have meaning in the context of whatever manufacturer you are talking about.
BB uses slim fit to identify their slimmer shirts. the term is used to distinguish from their traditional fit. for size 16, which I wear, traditional fit is something like 52.5 in the chest, while slim fit is 47.5 to 48 inches around in the chest (I'm guessing on the traditional fit, since I wear the slim fit). similar terms usually are tailored fit (used by Lands' end to indicate a shirt of similar dimensions as BB's slim fit), trim fit (used by Paul Fredrick), and fitted (used by Tyrwhitt, and slimmer than the three I just noted)
I think vintage fit is kind of a made up term - I think J. Crew uses vintage fit with their polo shirts, to indicate a tapered/tailored shirt. But I'm sure others use it to indicate baggier items, or just a shirt that's made off a "vintage pattern". I'm suspect classic fit is treated similarly.