Cleanup collaboration mode variants (#10404)

## Summary

This PR simplifies collaboration modes to the visible set `default |
plan`, while preserving backward compatibility for older partners that
may still send legacy mode
names.

Specifically:
- Renames the old Code behavior to **Default**.
- Keeps **Plan** as-is.
- Removes **Custom** mode behavior (fallbacks now resolve to Default).
- Keeps `PairProgramming` and `Execute` internally for compatibility
plumbing, while removing them from schema/API and UI visibility.
- Adds legacy input aliasing so older clients can still send old mode
names.

## What Changed

1. Mode enum and compatibility
- `ModeKind` now uses `Plan` + `Default` as active/public modes.
- `ModeKind::Default` deserialization accepts legacy values:
  - `code`
  - `pair_programming`
  - `execute`
  - `custom`
- `PairProgramming` and `Execute` variants remain in code but are hidden
from protocol/schema generation.
- `Custom` variant is removed; previous custom fallbacks now map to
`Default`.

2. Collaboration presets and templates
- Built-in presets now return only:
  - `Plan`
  - `Default`
- Template rename:
  - `core/templates/collaboration_mode/code.md` -> `default.md`
- `execute.md` and `pair_programming.md` remain on disk but are not
surfaced in visible preset lists.

3. TUI updates
- Updated user-facing naming and prompts from “Code” to “Default”.
- Updated mode-cycle and indicator behavior to reflect only visible
`Plan` and `Default`.
- Updated corresponding tests and snapshots.

4. request_user_input behavior
- `request_user_input` remains allowed only in `Plan` mode.
- Rejection messaging now consistently treats non-plan modes as
`Default`.

5. Schemas
- Regenerated config and app-server schemas.
- Public schema types now advertise mode values as:
  - `plan`
  - `default`

## Backward Compatibility Notes

- Incoming legacy mode names (`code`, `pair_programming`, `execute`,
`custom`) are accepted and coerced to `default`.
- Outgoing/public schema surfaces intentionally expose only `plan |
default`.
- This allows tolerant ingestion of older partner payloads while
standardizing new integrations on the reduced mode set.

## Codex author
`codex fork 019c1fae-693b-7840-b16e-9ad38ea0bd00`

codex-rs/protocol/src/config_types.rs

@@ -173,10 +173,23 @@ pub enum AltScreenMode {
 pub enum ModeKind {
     Plan,
     #[default]
-    Code,
+    #[serde(
+        alias = "code",
+        alias = "pair_programming",
+        alias = "execute",
+        alias = "custom"
+    )]
+    Default,
+    #[doc(hidden)]
+    #[serde(skip_serializing, skip_deserializing)]
+    #[schemars(skip)]
+    #[ts(skip)]
     PairProgramming,
+    #[doc(hidden)]
+    #[serde(skip_serializing, skip_deserializing)]
+    #[schemars(skip)]
+    #[ts(skip)]
     Execute,
-    Custom,
 }
 
 /// Collaboration mode for a Codex session.
@@ -276,7 +289,7 @@ mod tests {
     #[test]
     fn apply_mask_can_clear_optional_fields() {
         let mode = CollaborationMode {
-            mode: ModeKind::Code,
+            mode: ModeKind::Default,
             settings: Settings {
                 model: "gpt-5.2-codex".to_string(),
                 reasoning_effort: Some(ReasoningEffort::High),
@@ -292,7 +305,7 @@ mod tests {
         };
 
         let expected = CollaborationMode {
-            mode: ModeKind::Code,
+            mode: ModeKind::Default,
             settings: Settings {
                 model: "gpt-5.2-codex".to_string(),
                 reasoning_effort: None,
@@ -301,4 +314,13 @@ mod tests {
         };
         assert_eq!(expected, mode.apply_mask(&mask));
     }
+
+    #[test]
+    fn mode_kind_deserializes_alias_values_to_default() {
+        for alias in ["code", "pair_programming", "execute", "custom"] {
+            let json = format!("\"{alias}\"");
+            let mode: ModeKind = serde_json::from_str(&json).expect("deserialize mode");
+            assert_eq!(ModeKind::Default, mode);
+        }
+    }
 }