Salvages and supersedes #40163 (by @pinguarmy / 郝鹏宇). Cherry-picked onto current main with authorship preserved; all review feedback from that thread was already addressed by the author's latest commits, verified here against a clean rebase.

Summary

• Keep Desktop cached-session runtime state in sync with the active chat's model/provider/reasoning/fast/yolo fields.
• Restore persisted model/provider/reasoning/service-tier runtime when resuming stored sessions, instead of inheriting the latest global model from another chat.
• Persist live model/reasoning/fast changes back to the session row so future resumes keep the chat's latest runtime.
• Persist/restore custom endpoint base_url + api_mode in model_config on resume (raw api_key deliberately not stored in the session DB).
• Include provider in session.info and emit session.info after reasoning/fast/model changes.
• Add shared formatting for the footer/model label and expose reasoning/fast state in the model menu.

Root cause

Desktop had three stale-state paths:

1. The renderer cached one set of footer atoms globally, so switching sessions could keep showing another chat's runtime state.
2. session.resume rebuilt the backend agent from current global config rather than the stored session row, so an older chat whose DB row said gpt-5.4 + high reasoning could resume as the latest global gpt-5.5 + medium reasoning.
3. Live runtime changes made inside a session (model switch, reasoning effort, fast/service tier) were emitted to the current frontend but not consistently persisted for the next resume.

Review feedback addressed (from #40163)

• _apply_model_switch keeps the two-arg _restart_slash_worker(sid, session) and adds _persist_live_session_runtime(session).
• use-session-state-cache.ts adds the per-session runtime setters alongside the sameMessageList(...) reference-equality guard, not in place of it.
• Live-runtime persistence collapsed to a single update_session_meta(..., model) DB write (update_session_model only as a legacy fallback).
• Custom endpoint base_url/api_mode persisted/restored in model_config.
• session["model_override"] repopulated after a DB-restored resume so a later /new rebuilds from the same restored source of truth.

Test plan

• scripts/run_tests.sh tests/test_tui_gateway_server.py → 255 passed, 1 failed. The failure (test_browser_manage_connect_default_local_reports_launch_hint) reproduces identically on clean main in this local macOS env (a Chromium browser is installed, so the "not found" hint never fires) and is unrelated to this diff.
• npm run typecheck --workspace apps/desktop → passed.
• apps/desktop vitest: use-model-controls.test.tsx, model-status-label.test.ts, store/session.test.ts → 24 passed.

Closes #40163.