The diminished 7th in diminished 7th chords is a 6th, but it helps knowing it's a 7th and spelling it as such when you need to use this chord. They are a tool for modulation, because each note in the diminished 7th chord can be a diminished chord and be used to go to any key where that tone is a diminished chord in a scale.

Take a B diminished 7th, B, D, F, Ab. I would have been in the key of C major. My options of the next key I can go to are Eb major, using D - F - Ab. Or F# major, using F - Ab - B. Or A major, using Ab (enharmonic G#) G# - B - D. Makes those far away modulations easy. C major to F# major is pretty far away.

As for the minor 6th chords, they are a mess. I agree and thought the same thing when I first heard of them. In the the spelling the "minor or major" refers to as the triad, not the interval from root to sixth. However some people write them is different, I've seen Am6 for A, C, E, F. Then seen Am#6 for A, C, E, F#. That makes more sense to me, but then again everybody does it different. Chris is right though, when you see a Am6 it usually means A, E, C, F#. Because A, E, C, and F is really an inverted an Fmaj7 chord. F, A, C, E but with A in the bass.