Move diagnostics into a namespace (#5173)

What this really does is avoids shadowing names, so that we can
comfortable have things like `Check::DiagnosticEmitter` or
`Check::DiagnosticLoc` without shadowing being a concern.

Note, down this path I'm also thinking about:

- Renaming misc DiagnosticConsumer/DiagnosticEmitter classes, possibly
just to DiagnosticConsumer/DiagnosticEmitter (so
`Check::DiagnosticEmitter` instead of `SemIRLocDiagnosticEmitter`).
- Dropping `Diagnostic` from `Emitter::DiagnosticBuilder`.
- But not for `Check::DiagnosticBuilder`, because `Check::Builder` would
be ambiguous.
- Renaming diagnostics/diagnostic_* to drop "diagnostic".

[Discussion about SemIRLoc ->
DiagnosticLoc](https://discord.com/channels/655572317891461132/655578254970716160/1353771570463768698)
reminded me of this (in particular the older [Check::DiagnosticBuilder
discussion](https://discord.com/channels/655572317891461132/655578254970716160/1344363562608627763)),
but I'd only do that rename if there's matching consensus about a path
forward where we keep SemIRLoc, and in a way that it's only ever used
for diagnostics (the divergence from which is at the root of current
LocId discussion).

I'm trying to keep that separate from a namespace addition for clarity.

toolchain/language_server/language_server.cpp

@@ -44,7 +44,7 @@ class Logger : public clang::clangd::Logger {
 
 auto Run(FILE* input_stream, llvm::raw_ostream& output_stream,
          llvm::raw_ostream& error_stream, llvm::raw_ostream* vlog_stream,
-         DiagnosticConsumer& consumer) -> bool {
+         Diagnostics::Consumer& consumer) -> bool {
   // The language server internally uses diagnostics for logging issues, but the
   // clangd parts have their own logging system. We intercept that here.
   Logger logger(&error_stream, vlog_stream);
@@ -65,7 +65,7 @@ auto Run(FILE* input_stream, llvm::raw_ostream& output_stream,
     RawStringOstream out;
     out << err;
     CARBON_DIAGNOSTIC(LanguageServerTransportError, Error, "{0}", std::string);
-    NoLocDiagnosticEmitter emitter(&consumer);
+    Diagnostics::NoLocEmitter emitter(&consumer);
     emitter.Emit(LanguageServerTransportError, out.TakeStr());
     return false;
   }