Bonjour Stuart,

First of all, the verb "demeurer" is actually pretty rare in speech nowadays, as it's quite old-fashioned and 'posh', but you can still stumble upon it in written texts (novels, articles...).
In the present tense, it can indeed have both meanings, depending on context:
e.g. Mon ami demeure à Paris pour le moment. (My friend lives in Paris for now.)
Je demeure imperméable à ces attaques. (I remain impervious to his attacks.)

Note that these two meanings are actually very close to each other, in that they both contain the idea of permanence, of "staying", whether staying somewhere, or staying in a specific state of mind.

I hope that's helpful!
A bientôt !