FEAT-#7102: Deprecate enable_api_only mode in modin logging

[arunjose696]

What do these changes do?

• first commit message and PR title follow format outlined here

  NOTE: If you edit the PR title to match this format, you need to add another commit (even if it's empty) or amend your last commit for the CI job that checks the PR title to pick up the new PR title.

• passes flake8 modin/ asv_bench/benchmarks scripts/doc_checker.py
• passes black --check modin/ asv_bench/benchmarks scripts/doc_checker.py
• signed commit with git commit -s
• Resolves Remove enable_api_only mode in modin logging #7102?
• tests added and passing
• module layout described at docs/development/architecture.rst is up-to-date

[YarShev]

@arunjose696, please rebase on current master to fix CI.

[arunjose696]

@arunjose696, please rebase on current master to fix CI.

I have rebased on master, and have updated code just to add a deprecation message instead of removing the option.