Context mostly. But I've also seen a word other people have answered this question with that I haven't learned yet, but I'm pretty sure it's "skall" or 'skjall' - which only implies the future tense and not desire. I imagine it's pronounced like English "shall" but the [a] is not 'ae' digraph.
I saw a reply on a different exercise that said "vil" usually means "want to" for cases when a person has a choice (or "free will") whereas when there's no choice involved it means "will". I think an example like "the sun will come out today" because the sun (itself) can't choose whether or not to shine.
I think you're right about "skal" - it does mean "shall" and that's a good way of saying one intends to do something, whereas in English "shall" has become archaic and replaced with "will" at this point.
"Shall" is far from archaic, from a linguistic standpoint. It is still used with regularity in both speech and writing, and is fully understood by everybody. Perhaps "dated" is a more accurate description. (Though I would be personally inclined to contest that as well.)
Yes. They will go on (continue) to study = De skal fortsette å studere. They want to continue to study = De vil fortsette å studere.
"vil" is a false friend. It doesn't mean "will", but rather "want to".
Thank you. It's still sounding strange for me, because I intend "will" as an expression of will, so I wrote "they will go on" meaning "they will go on because they want to go on" (this ambiguity between "will" expressing a free will and "will" expressing future is allowed in English – as is in Norwegian –, while "shall" is more expressing a duty). It's strange for me to think to translate "will" with "skal".
Yes. But when you hear Norwegians talking, they say "skal" to basically everything they are going to do. "Skal" does not only translate to "shall", but also "must", "should", "will", "going to".
However, as the first video explains: You use "vil" when you have no (direct) influence on the action, because it is either just a wish or if it does not depend on you (because you need permission or something).
So if you express something you are going to do: Use "skal". - Jeg skal på butikken fordi jeg skal kjøpe mat.
If you express something you wish to do: Use "vil". - Jeg vil på ferie, men jeg må jobbe.