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/check/modifiers.cpp

@@ -11,9 +11,10 @@ namespace Carbon::Check {
 // Builds the diagnostic for DiagnoseNotAllowed.
 template <typename... TokenKinds>
 static auto StartDiagnoseNotAllowed(
-    Context& context, const DiagnosticBase<TokenKinds...>& diagnostic_base,
+    Context& context,
+    const Diagnostics::DiagnosticBase<TokenKinds...>& diagnostic_base,
     Parse::NodeId modifier_node, Lex::TokenKind declaration_kind)
-    -> DiagnosticEmitter<SemIRLoc>::DiagnosticBuilder {
+    -> Diagnostics::Emitter<SemIRLoc>::DiagnosticBuilder {
   if constexpr (sizeof...(TokenKinds) == 0) {
     return context.emitter().Build(modifier_node, diagnostic_base);
   } else if constexpr (sizeof...(TokenKinds) == 1) {
@@ -34,7 +35,8 @@ static auto StartDiagnoseNotAllowed(
 // declaration kind.
 template <typename... TokenKinds>
 static auto DiagnoseNotAllowed(
-    Context& context, const DiagnosticBase<TokenKinds...>& diagnostic_base,
+    Context& context,
+    const Diagnostics::DiagnosticBase<TokenKinds...>& diagnostic_base,
     Parse::NodeId modifier_node, Lex::TokenKind decl_kind,
     SemIR::LocId context_loc_id) -> void {
   auto diag = StartDiagnoseNotAllowed(context, diagnostic_base, modifier_node,