[Bug] Session name persists in status bar after /clear command

Bug Description
---
Bug Report

Title: [BUG] Session name persists in status bar after /clear — shows previous session's name

Body:

### Preflight Checklist

• [x] I have searched existing issues and confirmed this is not a duplicate
• [x] I can reproduce this bug consistently

### Bug Description

After running /clear, the session name label in the bottom-right of the status
bar still displays the previous session's name instead of resetting. This is
confusing because /clear creates a new session (new .jsonl file), but the UI
still shows the old session's identity.

### Reproduction Steps

1. Start claude in interactive mode
2. Name the session (or let it auto-name), e.g., "context7-library-docs"
3. Run /clear
4. Observe the status bar (bottom-right corner) — it still shows

"context7-library-docs" even though a new session file has been created

### Expected Behavior

After /clear, the session name in the status bar should either:

• Reset to empty/default (no name chip shown)
• Show the new session's name (or "unnamed")

### Actual Behavior

The old session name persists in the status bar, making it appear
you're still in the same session when you're actually in a new one.

### Environment

• Claude Code version: (run claude --version)
• OS: macOS Darwin 25.1.0
• Terminal: (your terminal app)
• Model: Claude Opus 4.6

### Related Issues

• #16353 — Similar but about terminal tab title, not status bar session name
• #13765 — Context window tracking not resetting on /clear
• #9352 — /clear breaks session continuity (closed)

### Screenshots

(Attach your screenshot showing the stale session name after /clear)

---

Environment Info

• Platform: darwin
• Terminal: Apple_Terminal
• Version: 2.1.42
• Feedback ID: 28405e5c-4bc6-4058-a021-8cba9ffe0951

Errors

[{"error":"Error: NON-FATAL: Lock acquisition failed for /Users/digora/.local/share/claude/versions/2.1.42 (expected in multi-process scenarios)\n    at TzA (/$bunfs/root/claude:3483:2098)\n    at ChL (/$bunfs/root/claude:3483:1209)\n    at processTicksAndRejections (native:7:39)","timestamp":"2026-02-16T10:48:52.156Z"}]