What does “living” mean—is it being alive, or is it enjoying what you’re doing?

The term “living” should refer to enjoying what you’re doing; at least, that is how I use it.

Example: That lady playing on the jungle gym is truly living her best life.

Yes, it could mean being alive as well , it just depends on how you use it in your sentence.  Example: Yes, my aunt is still living at the age of 97.