If people want to set up life in a new place...
They tend to 'make it their own' by naming it themselves....
So I doubt that inhabitants would call the moon 'the moon'- at least not after the first generation, they would probably consider the term demeaning- it indicates that 'the moon' is secondary... With Earth being primary.
As for Mars- well thats not as simple- but eventually they would develop their own name from their planet, or thgeir own region of the planet...
We were not calling our country the USA at first, or at second.... The ottoman empire is an outdated name....
Names are fluid- they change over history...
If the Chinese get to mars first, I have no idea what they will call it- they probably wont name it Mars or Aries...