In a way, yes (should be accepted). Hiihtäminen sounds like the physical act of skiing, whereas hiihto refers to skiing as a sport.
Hiihtäminen on vaikeaa (Skiing is difficult) vs. Hiihto on suosittu urheilulaji Suomessa (Skiing is a popular type of sport in Finland)
There are some overlaps: I can imagine both Hiihtäminen/hiihto on mukavaa (Skiing is enjoyable) depending on which meaning you're leaning towards.
But! Murtomaahiihto (cross-country skiing) & lasketteluhiihto (downhill skiing) etc. are TYPES of skiing so using hiihtäminen would be weird.
I completely agree with most of your post. However, I'm not sure if I understand the last paragraph.
I would never use hiihtäminen for laskettelu (I believe downhill skiing doesn't have hiihto in it?) because even the verb is different: lasketella. However, I would certainly use hiihtäminen for doing cross-country skiing. So... I think I kind of can't see the point. But other than that, I'm with you.
Maybe according to Wikipedia. But for us Finns that actually uses these words, this just doesn't make sense. If you speak about just skiing the word is hiihto and doing it(verb) IS hiihtää/hiihtäminen. But if the thing you are after is referring to the actual singural sport it is called murtomaa(cross-country) skiing(hiihto).
Sure it does. Practically all of us Finns will automatically understand hiihto to mean murtomaahiihto. Our normal activity during the winter.
For an English-speaking person skiing doesn't mean that at all. They'll think of laskettelu, and usually cross-country skiing has to be explained to them, because it's quite rare compared to downhill skiing.
In general Duo refers (in this and other exercises) to the activity of moving yourself with just skis and poles (simply put), which we Finns most commonly call hiihto, but also murtomaahiihto and maastohiihto. In English, all those are just cross-country skiing. It doesn't translate properly to just skiing, because it has a different meaning. And it's also not just about an actual singular sport as such, but the activity in general, casual or competitive, ladulla tai hangessa.