My explanation why Amy is called "the GIRL who waited" is that she was just a girl when she first met the Doctor and she was sort of still a child in her mind when the Doctor returned after 14 years. (Does that make sense?) But I don't really understand why Clara is also titled as girl. She doesnt necessarily need the Doctor, she is quite confident with her life as adult. So why do they call her impossible girl rather than impossible women?
Because the primary intended viewership is in fact families and children.
This is something that Moffat has shown a greater awareness of than any other New Who writer. For instance, his episodes make up the grand majority of those which have the Doctor interacting with children, from Reinette Poisson and CAL during the RTD era to little Amelia Pond, Kazran and others in his time as a showrunner. And you can see it in his use of language as well - characters of any gender are more likely to be refered as “boys” or “girls”.
Clara and her monikers are part of that. She is, by the way, not always called a girl, she’s also “the woman twice dead” and the first word the Doctor uses to describe her when he realises that there is something mysterious about her is “woman” as well (“The same woman, twice!”). But, yes, overall she’s much more likely to be refered to as a “girl” than a “woman.
And I think that is absolutely alright.
If this fandom in general could maybe stop viewing “girl” as a bad word just about now, I really wouldn’t mind. Because, frankly, the only way one would be able to hold up this definition would be to actually buy into the negative connotation which society has attached to the term and decide not to challenge this at all (and moreover shoot down any attempts to do so by others).
Because challenging this is what calling Clara “The Impossible Girl” does. It doesn’t belittle her, because as you point out, she is confident, mature, well-adjusted and just fine on her own and her moniker has no impact on that. Instead it takes a wonderful character like that and makes a point to say to all the young girls watching, “this is someone just like you”.
And that’s rather important, I think.