fix(runs): persist tmux-backed runs so a resumed session shows its own info

`dashboard_runs.mode` is NOT NULL and `recordRun` swallows insert errors by
design, so pty-run.js passing `mode: null` meant NO tmux-backed run was ever
persisted — the newest row in a live database dated from before pty-run.js
replaced run-spawner.js. `publicRun` reads cwd, model, prompt preview,
started_at and session_id off that row and the handle itself is recomputed
from tmux, so the runs modal rendered a live row with no cwd, no model, no
prompt and a start date of Jan 1 1970. The same missing row is why a resumed
session kept its separate `external` badge: the client dedupes an external
session against the session ids carried by run rows, and no run row ever
carried one.

- pty-run.js records `mode: "interactive"` — every run here is one pane.
- dashboard-runs.js still never fails a live run on a db error, but warns
  instead of swallowing silently; that silence hid this for six weeks.
- RunHistory renders `startedAt: 0` as "-" rather than dating a row with no
  known start time to the epoch.

Also closes the other half of the same lie: adopting a pane that is already
running an agent now throws ERUNLIVE (409 from both the lane start action and
POST /api/run) when the request carried a prompt or a resumeSessionId. That
argv cannot be typed over a live agent, so answering 200 reported a resume
that never spawned. A start asking for nothing but a pane still adopts.

docs/API.md

@@ -245,7 +245,7 @@ All fields are optional. The tool (CLI invoked) comes from the lane itself, neve
 - **`approval`** — one of `on-request` or `never`, or omitted/empty for no approval prompt. Passed to `codex -a <value>`.
 - **`resumeSessionId`** or **`resumeLast: true`** — resume an earlier run instead of starting new. For Codex lanes specifically, `resumeLast: true` uses `codex resume --last`; `resumeSessionId` uses `codex resume <sessionId>`. For Claude lanes, `resumeSessionId` re-attaches a Claude session by id.
 
-Error `409 ERUNLIVE` is returned if the lane already has a live run; stop it first with the `stop` action.
+Error `409 ERUNLIVE` is returned if the lane already has a live run; stop it first with the `stop` action. The same code and status come back when the lane's tmux pane is already running an agent while a `prompt` or `resumeSessionId` was sent — that argv cannot be typed over a live agent, so the request is refused instead of answered 200 with nothing done. A start that asks for neither still adopts the existing pane silently.
 
 #### Preflight a destructive lane action