Replace #16455
Close #21803
Mixing different Gitea contexts together causes some problems:
1. Unable to respond proper content when error occurs, eg: Web should
respond HTML while API should respond JSON
2. Unclear dependency, eg: it's unclear when Context is used in
APIContext, which fields should be initialized, which methods are
necessary.
To make things clear, this PR introduces a Base context, it only
provides basic Req/Resp/Data features.
This PR mainly moves code. There are still many legacy problems and
TODOs in code, leave unrelated changes to future PRs.
