Use DiagnosticEmitter for phase-specific types (#5188)

Given the namespacing of `Diagnostics` in #5173, now we can use
`DiagnosticEmitter` for phase-specific emitters. This is consistent with
how we do `Context`, and also check had started this with
`DiagnosticBuilder` in anticipation of the namespacing.

Also renames `Emitter::DiagnosticBuilder` to `Emitter::Builder` for
consistency with other `Diagnostics` entities.

In check, I'm still splitting `DiagnosticEmitterBase` and
`DiagnosticEmitter` just to keep the emitter definition separate from
the context.

Also cleans up some incorrect check diagnostic emitter dependencies in
lower.

toolchain/check/modifiers.cpp

@@ -14,7 +14,7 @@ static auto StartDiagnoseNotAllowed(
     Context& context,
     const Diagnostics::DiagnosticBase<TokenKinds...>& diagnostic_base,
     Parse::NodeId modifier_node, Lex::TokenKind declaration_kind)
-    -> Diagnostics::Emitter<SemIRLoc>::DiagnosticBuilder {
+    -> DiagnosticBuilder {
   if constexpr (sizeof...(TokenKinds) == 0) {
     return context.emitter().Build(modifier_node, diagnostic_base);
   } else if constexpr (sizeof...(TokenKinds) == 1) {