Investigate options for logging LanguageServer stderr/stout/exit codes

We don't have anything from the logs captured for #10113 to figure out the root cause. Having the stderr + exit code might enable us to better troubleshoot activation failures.

No files, tests, or entry points are named. Start by tracing how LanguageServer activation failures are currently logged and compare the available information with the failure reported in #10113. Done means documenting viable options for capturing stderr, stdout, and exit codes so activation failures can be investigated.

Written by the indexing model from the issue text.