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/eval.cpp

@@ -187,9 +187,7 @@ class EvalContext {
 
   auto sem_ir() -> SemIR::File& { return context().sem_ir(); }
 
-  auto emitter() -> Diagnostics::Emitter<SemIRLoc>& {
-    return context().emitter();
-  }
+  auto emitter() -> DiagnosticEmitterBase& { return context().emitter(); }
 
  private:
   // The type-checking context in which we're performing evaluation.