iOS app shows a session's name as "2" instead of its title

**What happens**

A local session titled "Basalt" on the macOS desktop app appears as `2` in the iOS app's Code tab.

**Why it looks like a display bug rather than a sync one**

Renaming the session on the desktop does not change the iOS entry, but its activity timestamp updates live in the same refresh ("22m" → "now"). So the iOS app is receiving updates from that exact session and is rendering the name from something other than the session title.

Confirmed by changing the title on the desktop twice ("Basalt" → "Basalt platform" → "Basalt") and refreshing iOS after each: the timestamp tracked the session's activity every time, the name stayed `2` throughout.

**Steps to reproduce**

1. Desktop app: have a session with a real title, e.g. "Basalt"
2. iOS app, Code tab: the session is listed as `2`
3. Rename the session on the desktop
4. Pull to refresh on iOS — the timestamp updates, the name stays `2`

**Expected** — the session's title

**Actual** — `2`

**Environment**

- macOS desktop app, model Opus 5
- iOS Claude app, Code tab
- The session is a local desktop session, not a Remote Control session

**Possibly related**

Only currently-running sessions appear in the iOS list; idle ones are absent. That may well be intended, but it is worth confirming, since it makes the wrong name harder to work around — there is no way to rename a session from iOS, and renaming on the desktop does not propagate.

Reproduce with a local macOS desktop session that has a real title, then view it in the iOS app's Code tab and pull to refresh after renaming it. Start by tracing which session field the iOS Code tab displays versus the title updated by the desktop app. Done means the iOS entry shows the session title, not `2`, while activity timestamps still update.

索引モデルが issue の本文から書いたものです。