The analogy I heard was that canvas shoes with rubber soles were first invented in the 1830's, but this doesn't mean that mountain men wore sneakers.
I don't think you would have seen much, if any, damascus steel in the Rockies, back in the day. The cutlers supplying the fur trade did use shear steel, which I believe is steel with a carburized surface. To over simplify, this could be folded and laminated once to put the carburized metal in the middle of the blade, so if the blade was beveled equally on both sides, the carburized metal would be at the cutting edge.
A lot of old Indian knives were sharpened on just one side. There have been a lot of theories as to why this was done (better for skinning, better for slicing jerky, easier to sharpen...), but a buddy of mine suggested years ago that the really cheap trade knives were of single-shear steel, i.e. only carburized on the surface. If you sharpen the knife equally on both sides, you have soft metal at the cutting edge. Sharpening on only one side would leave some of the harder, carburized steel at the edge, for better wear. That seemed plausible to me.
Please don't be offended, but that knife on the left in the first picture looks more like some sort of Japanese fighting knife than anything you would have seen in the fur trade. It is true that there were "country made" knives in use... at least a couple were known to have been made by mountain man Jim Baker... but if you read the period literature, the terms you see used most are butcher knife, and maybe to a lesser extent, scalping knife. This is from Rufus Sage, Rocky Mountain Life
... and this is from George Frederick Ruxton, Adventures in Mexico and the Rocky Mountains
Stick or "rat tail" tangs were known on some trade knives, but I don't think they used much on the knives supplied to the Rocky Mountain fur trade. The majority of these had half-tangs, tapering in thickness, with slab or scale handles, like the three originals on the left in the image below:
Another thing you see mentioned in the period literature is "soft steel." These knives needed frequent sharpening. Note in the quote from Ruxton (above) the whetstone carried on the trapper's belt. Your butcher knife, on the right in your first photo, looks like a nice old knife; maybe not quite pre-1840, but it could easily pass for a 19th century design, which ought to be close enough. If it doesn't keep an edge well, it is probably even more period correct!
In any event, you can carry what you want, but the knife on the left in that first photo would be very atypical for a frontier trapper's equipment. Your old knife, on the right, is a much better choice. Make a plain, simple one-piece sheath for it of rawhide, and you'll be well equipped.