Decouple the different contexts from each other (#24786)

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.

routers/api/v1/user/helper.go

@@ -17,7 +17,7 @@ func GetUserByParamsName(ctx *context.APIContext, name string) *user_model.User
 	if err != nil {
 		if user_model.IsErrUserNotExist(err) {
 			if redirectUserID, err2 := user_model.LookupUserRedirect(username); err2 == nil {
-				context.RedirectToUser(ctx.Context, username, redirectUserID)
+				context.RedirectToUser(ctx.Base, username, redirectUserID)
 			} else {
 				ctx.NotFound("GetUserByName", err)
 			}