Remove first-party type from app metadata (#34844)

## What changed

Remove `first_party_type` from connector app metadata and the app-server v2
protocol. Update metadata conversion and merge paths, generated JSON and
TypeScript schemas, and affected test fixtures to match.

GitOrigin-RevId: 3f2e56eca68352360d8c0d99d48486ae65d664c1

codex-rs/tui/src/app_info.rs

@@ -100,7 +100,6 @@ fn app_metadata_from_api(metadata: ApiAppMetadata) -> AppMetadata {
         version,
         version_id,
         version_notes,
-        first_party_type,
         first_party_requires_install,
         show_in_composer_when_unlinked,
     } = metadata;
@@ -119,7 +118,6 @@ fn app_metadata_from_api(metadata: ApiAppMetadata) -> AppMetadata {
         version,
         version_id,
         version_notes,
-        first_party_type,
         first_party_requires_install,
         show_in_composer_when_unlinked,
     }