save checkpoints and profiler output to the first logger

[tshu-w]

What does this PR do?

Fixes #12177 save checkpoints and profiler output to the first logger, which is a blocker of #14188.

Does your PR introduce any breaking changes? If yes, please list them.

Yes, as discussed in #12177.

I also delete pytorch_lightning.utilities.logger._name directly without marking them as deprecated since it is not used in another place and it doesn't break an API.

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[codecov]

Codecov Report

Merging #14325 (14b398d) into master (70fe0ed) will decrease coverage by 3%.
The diff coverage is 100%.

@@            Coverage Diff             @@
##           master   #14325      +/-   ##
==========================================
- Coverage      79%      76%      -3%     
==========================================
  Files         111      332     +221     
  Lines        7258    26920   +19662     
==========================================
+ Hits         5740    20449   +14709     
- Misses       1518     6471    +4953