[clang-tools-extra] [clangd] Allow specifying what headers are always included via "" or <> (PR #67749)

[kleines Filmröllchen via cfe-commits]

kleinesfilmroellchen wrote:

> It's sad that there's not a single codepath that would handle all these, but maybe the perfect is the enemy of the good here.

The code duplication was my reason for looking at the common path. If you're fine with duplicating the code I will go ahead and do that.

I have updated the commit with the review comments and added end-to-end tests.

https://github.com/llvm/llvm-project/pull/67749