Log debug and informational messages to stderr only

What's the problem this feature will solve?

Commands that generate output for the user (pip list, pip freeze) must be concerned with debug output.

Describe the solution you'd like

As mentioned in #6099:

1. all log messages should go to stderr instead of stdout
2. command API related output should use a dedicated interface that outputs to stdout

Any issue that seeks to make the CLI more usable would benefit from this change. This also avoids issues like #6024.

Alternative Solutions

Originally, ad-hoc redirection of logger output to stderr was used to preserve command output. This could be used again, but it would make logging output from commands inconsistent in pip.

Additional Context

• #6099