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Many words in the English language have more than one meaning, according to the context. My favorite example is the word "scale."

In this case, within our own system the term "solar" refers to our own star. But when it is used for other stellar systems, it means "star" as such.

The key question, by the way, for proper word usage is if the particular usage in a case being examined is whether or not the usage would lead to confusion. In this case I think it would not.

Amen. I said exactly this yesterday on another post about this. If there is only one Sol, there is no "other Solar system." You'd think after decades of Star Wars/Trek, people would be accustomed to recognizing star systems, of which Sol is merely one of many.

Genuine question: f another star had a "solar flare", would the correct term actually be "stellar flare", or would solar be acceptable in that usage?

"Stellar flare" would indeed be correct. What would be more correct is "$adjective_form_name flare" for whatever that star is named.