Emit function definitions in check, for all specifics seen. (#5090)

Emitting definitions in check. This resolves the crash in lowering which
necessitated definitions be emitted.
Some of the test changes need further review.

toolchain/check/handle_class.cpp

@@ -275,7 +275,7 @@ static auto BuildClassDecl(Context& context, Parse::AnyClassDeclId node_id,
   }
 
   if (!is_definition && context.sem_ir().is_impl() && !is_extern) {
-    context.definitions_required().push_back(class_decl_id);
+    context.definitions_required_by_decl().push_back(class_decl_id);
   }
 
   return {class_decl.class_id, class_decl_id};