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/app-server-protocol/schema/json/ServerNotification.json

@@ -1129,7 +1129,7 @@
                   "$ref": "#/definitions/ModeKind"
                 }
               ],
-              "default": "code"
+              "default": "default"
             },
             "model_context_window": {
               "format": "int64",
@@ -3901,10 +3901,7 @@
       "description": "Initial collaboration mode to use when the TUI starts.",
       "enum": [
         "plan",
-        "code",
-        "pair_programming",
-        "execute",
-        "custom"
+        "default"
       ],
       "type": "string"
     },