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/context.cpp

@@ -8,7 +8,7 @@
 
 namespace Carbon::Check {
 
-Context::Context(Diagnostics::Emitter<SemIRLoc>* emitter,
+Context::Context(DiagnosticEmitterBase* emitter,
                  Parse::GetTreeAndSubtreesFn tree_and_subtrees_getter,
                  SemIR::File* sem_ir, int imported_ir_count, int total_ir_count,
                  llvm::raw_ostream* vlog_stream)