When is Research Assistant/Associate/Fellow a correct title for a PhD student? What is the correct term for a PhD student who does entirely research and is paid through third-party funds?

One can see that sometimes people doing a PhD call themselves a Research Associate or Research Assistant (in a very few cases also Research Fellow). I am looking for clarification regarding which term would be appropriate and also why the others are not.

I can't find a clear definition. If this is country dependent I would like to know this for the UK and Germany.
How about "PhD student"?

Since you also asked about Germany: in German you can call yourself "Doktorand" or "Promovend". But also in Germany, "PhD student" would be perfectly fine.

If you want to leave out the "student" part, you might call yourself "PhD candidate".

Be careful to avoid calling yourself something you are not (e.g. "Dr."), since particularly in Germany that might be illegal.