Neat find -
Many of the mountaineers learned to read via the "Rocky Mountain College" during winter camp.
The earliest usage of the term "mountain man" I've found was by a woman traveling west in the early 1840's - of course I can't find the exact cite now (hate when that happens!), but it was IIRC 1842-43 and she was a missionary lady.....but the term apparently didn't come into general usage until sometime later, after Fremont's trips and those first penny dreadfuls came out about Carson, Bridger, and others.....
Mountaineer and trapper seem to be the most often used terms by the men themselves and Rudolph Kurz, who clerked at Ft. Union, used those two terms as late as 1851-2.