I'm not sure about AASHTO, but with dimensioned lumber you are correct. Your 2x10 is 10" wide... actual lumber dimensions for a 2x10 are 1-1/2 x 9-1/4. The 2x10 is a nominal dimension, and not actual.
I'm confused with what "size classification" a 2x10 would be in AASHTO LRFD Table 8.4.1.1.4-1. That table gives different design values (allowable stresses prior to applying various strength factors) for the different size classifications which I listed in the original post.
Depends on how the lumber is oriented. "Wide" is the horizontal dimension. "Thick" is the vertical dimension. A 2x10 used as a typical beam is 2" wide.
A 2x10 used as, say, deck flooring is 10" wide.
Take a look at the attached, PS-20, The American Softwood Lumber Standard. I believe this is what AASHTO follows. Also, look at the Commentary to AASHTO 8.4.1.1.1
My screw up... please ignore my comment. The 2" nominal dimension is the width and 10" would be the depth... don't know what I was thinking when I replied.