Refactor web package and context package (#25298)

1. The "web" package shouldn't depends on "modules/context" package,
instead, let each "web context" register themselves to the "web"
package.
2. The old Init/Free doesn't make sense, so simplify it
* The ctx in "Init(ctx)" is never used, and shouldn't be used that way
* The "Free" is never called and shouldn't be called because the SSPI
instance is shared

---------

Co-authored-by: Giteabot <teabot@gitea.io>

modules/context/api.go

@@ -20,6 +20,8 @@ import (
 	"code.gitea.io/gitea/modules/httpcache"
 	"code.gitea.io/gitea/modules/log"
 	"code.gitea.io/gitea/modules/setting"
+	"code.gitea.io/gitea/modules/web"
+	web_types "code.gitea.io/gitea/modules/web/types"
 
 	"gitea.com/go-chi/cache"
 )
@@ -41,6 +43,12 @@ type APIContext struct {
 	Package *Package
 }
 
+func init() {
+	web.RegisterResponseStatusProvider[*APIContext](func(req *http.Request) web_types.ResponseStatusProvider {
+		return req.Context().Value(apiContextKey).(*APIContext)
+	})
+}
+
 // Currently, we have the following common fields in error response:
 // * message: the message for end users (it shouldn't be used for error type detection)
 //            if we need to indicate some errors, we should introduce some new fields like ErrorCode or ErrorType