[clang] [clang] Apply internal buffering to clang diagnostics printing (PR #113440)

[Mariya Podchishchaeva via cfe-commits]

Fznamznon wrote:

BTW, the test failure IS caused by the patch. It somehow caused permanent swap of what clang-tidy prints in clang-tidy-run-with-database.cpp test. Not sure why...

https://github.com/llvm/llvm-project/pull/113440