[TASK] Improve error handling for API v1 on argilla-server

[jfcalvo]

The idea of this issue is to improve how we are managing error handling on API v1 and future versions.

There is an first exploration of an improvement on this PR: https://github.com/argilla-io/argilla/pull/4856

Improvements that should be done:

• We should map custom Argilla exceptions to the correct response on FastAPI so try..except blocks are not necessary.
• We should avoid to use old API v0 exceptions on API v1.
• We should return a simpler response and always include an error code (like we are doing on PR #4856).