Use stdout for error messages

[tmat]

Messages written to stdout and stderr are not guaranteed to be displayed in the order they were written. This might cause confusion as an error might not be shown before we prompt the user for action, etc. It also causes intermittent failures of tests that verify error messages.

[tmat]

@phil-allen-msft ptal

[dotnet-policy-service]

Thanks for your PR, @tmat.
To learn about the PR process and branching schedule of this repo, please take a look at the SDK PR Guide.

[KalleOlaviNiemitalo]

stderr design in #7289

[tmat]

@KalleOlaviNiemitalo
dotnet-watch outputs 4 different categories of output: verbose, ordinary, warning, error. All are regular, expected output. They vary in significance to the user. There are no scenarios where it would be useful to run dotnet-watch and only see errors, or non-errors. On the other hand, a common scenario is to display verbose messages (--verbose), which are not printed out by default.

Ordinary output describes context for error messages. It is important that both are printed out in the correct order, otherwise the proper context might be missing.