[libcxx] [llvm] [libc++][C++03] Use `__cxx03/` headers in C++03 mode (PR #109002)

[Louis Dionne via llvm-commits]

https://github.com/ldionne approved this pull request.

I'm going to approve this with changes requested, and assuming CI is green. This should unblock you to keep moving forward over the holidays. I'd really like us to iterate in a separate branch until we gain a bit more confidence.

https://github.com/llvm/llvm-project/pull/109002