Remove TS{####} errors from --watch CLI

[JoshuaKGoldberg]

After #20416, this is what the CLI will look like (and without it, it's the same but with fewer colors):

AFAIK the error codes like TS2322 are really only used in Visual Studio, so folks using --watch wouldn't need them. Can we just remove them for being visual clutter?

[mhegazy]

This debate goes back to 1.1 time-frame, (see #694). I think we have agreed to keep them for now, as useless as they may be.