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/tui/src/chatwidget/settings.rs

@@ -605,7 +605,7 @@ impl ChatWidget {
         }
         match self.active_mode_kind() {
             ModeKind::Plan => Some(CollaborationModeIndicator::Plan),
-            ModeKind::Default | ModeKind::PairProgramming | ModeKind::Execute => None,
+            ModeKind::Default => None,
         }
     }