agree with Colin but breaking it down further,
for example,
In C major the 6th note of the scale is A but in reality to get to it using all interval notes it is 9 chromatic steps as you said (c,c sharp,d,d sharp,e, f, f sharp, g, g sharp then A). this is chromatically tho....that is where you are getting mixed up by thinking of it in the literal sense moving up chromatically instead of in the sense i outline below.
The term "major" 6th has a clue in the title as it relates to the steps needed to get up to it in a MAJOR scale, hence major 6th, so if we go up in steps using the formula for the C major scale it is 6 steps - C,D,E,F,G,A.
this applies to all related major 6th intervals in all major keys.
using the key signatures of the starting note of the major key.
D the major 6th is B - d,e,f sharp, g, a then b.
Eb the major 6th is C. - Eb, f,g,Ab, Bb then c.
and so on