Replace unnecessary `gameclient.h` include with more specific includes.
Fix storage creation error message not being logged as the logger was initialized after checking for the failed storage creation. However, in this case we want to avoid non-error log messages so the tool's output is only the extracted demo chat, except in error cases.
Rename `Process` function to `ExtractDemoChat` and make it `static` to avoid exporting it.
Use `log_error` instead of `dbg_msg`.