minor: Show build scripts errors in server status

[alibektas]

Not the prettiest PR so I want to suggest a few things : ( fixes #19184 )

1. While I was tinkering with our build script runners, I saw that we ignore any Warn level diagnostics but if there are any errors, warnings would also be included in the "problems". Since the PR simply adds showing what cargo check outputs as a server status message, users will now also see these warning when there are errors and not see them where there are not. This tells me that we can add a DiagnosticLevel guard to filter warnings out.
2. I could better format error messages, since they are not very pleasant to look at.