The majority of puzzles I have found are solve puzzle, usually from home, then go and find cache at co-ords you have generated, occasionally there is more to do, a twist or two, for the puzzles I have found these are usually only puzzles from blindmouse though, almost expected that there will be a twist to be honest
Occasionally I miss puzzles that are listed as puzzles, but are actually go to listed co-ords, find a sign, do some work to build co-ords and find cache, personally I see these as multi caches, so I agree there is some cross over on types, but again I think these types are more caches of the older variety, I find the newer ones, post 2010, seem to be set as the 'correct' type (or 'correct' as I understand each cache type)
I certainly wouldn't have them as one cache type, in a nutshell I see Puzzle as 'work required from home' and multi 'I can do this with no research beforehand' the majority of the time this rings true, but there are always exceptions....
The majority of Letterbox caches I have done are only that type due to there being a stamp in the box, I can think of two where extra work was required, giving them a puzzle element, but using what I understand to be Letterbox techniques, these two are listed below:
CTC - Burial Lane (Letterbox Hybrid) (GC19M60)Stagecoach Stash (GC136H4)This one was a little different, as the cache hide was a small red post box container, inside an actual letterbox that had been attached to a fence
YO7 56 Thorpefields (GC30N1C)