Track TUI starts by app server mode (#41974)

## What changed

Emit the `codex.tui.start` counter once per TUI launch with an
`app_server_mode` label of `in_process`, `local_daemon`, or `remote`. Recording
the metric during startup keeps reconnects from counting as additional
launches.

GitOrigin-RevId: 7527f319b95ae53f5278fa16120120a433a29ae4

codex-rs/tui/src/startup_orchestration.rs

@@ -400,6 +400,17 @@ pub(super) async fn run_main_inner(
     };
     if let Some(metrics) = otel.as_ref().and_then(codex_otel::OtelProvider::metrics) {
         let _ = codex_otel::record_process_start_once(metrics, otel_originator.as_str());
+        // Count the selected mode once per TUI launch, independently of reconnects.
+        let app_server_mode = match &app_server_target {
+            AppServerTarget::Embedded => "in_process",
+            AppServerTarget::LocalDaemon { .. } => "local_daemon",
+            AppServerTarget::Remote { .. } => "remote",
+        };
+        let _ = metrics.counter(
+            "codex.tui.start",
+            /*inc*/ 1,
+            &[("app_server_mode", app_server_mode)],
+        );
         let telemetry =
             codex_rollout::sqlite_telemetry_recorder(metrics.clone(), otel_originator.as_str());
         let _ = codex_state::install_process_db_telemetry(telemetry);