Remove legacy collaboration mode variants (#36990)

## What changed

- Remove the hidden `PairProgramming` and `Execute` variants from `ModeKind`.
- Delete their unused prompt templates and simplify mode handling to cover only `Default` and `Plan`.

GitOrigin-RevId: 8d24e3f562a09e3071b2decae275d2b3871a1e9c

codex-rs/core/tests/suite/request_user_input.rs

@@ -411,19 +411,6 @@ where
     Ok(())
 }
 
-#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
-async fn request_user_input_rejected_in_execute_mode_alias() -> anyhow::Result<()> {
-    assert_request_user_input_rejected("Execute", |model| CollaborationMode {
-        mode: ModeKind::Execute,
-        settings: Settings {
-            model,
-            reasoning_effort: None,
-            developer_instructions: None,
-        },
-    })
-    .await
-}
-
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
 async fn request_user_input_rejected_in_default_mode_by_default() -> anyhow::Result<()> {
     assert_request_user_input_rejected("Default", |model| CollaborationMode {
@@ -441,16 +428,3 @@ async fn request_user_input_rejected_in_default_mode_by_default() -> anyhow::Res
 async fn request_user_input_round_trip_in_default_mode_with_feature() -> anyhow::Result<()> {
     request_user_input_round_trip_for_mode(ModeKind::Default).await
 }
-
-#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
-async fn request_user_input_rejected_in_pair_mode_alias() -> anyhow::Result<()> {
-    assert_request_user_input_rejected("Pair Programming", |model| CollaborationMode {
-        mode: ModeKind::PairProgramming,
-        settings: Settings {
-            model,
-            reasoning_effort: None,
-            developer_instructions: None,
-        },
-    })
-    .await
-}