Sounds like you're thinking of 6 as in figured bass which is indeed shorthand for 1st inversion (6 3 1 chord structure going downwards).
I'd have thought add6 means just a sixth added to a major or minor triad, whilst 13 implies 11, 9 and 7 are also included in the chord. Different beasts altogether.
Min add 6 is such a distinctive spicy chord that it tends (in my rip off world of media music) to only get used for soundalikes of James Bond theme, 'that' Guinness ad tune, Summertime and perhaps cheesy swing style Big Band sax quartets moving between Maj add6 and dim7 chords.
Have to admit though, in the real world, something like a dom 9th chord which actually has no 3rd eg. C,Bb,D,G (going upwards) would often be referred to as a 9th chord in the context of say, a James Brownesque funk guitar riff. Of course in theory world however, it snot though....