feat(desktop): send meeting transcript segments and preset prompt edits to the phone

The phone draws a meeting from its transcript segments before the edited
transcript, so desktop edits, auto-polish and diarization never showed there.
Every desktop transcript change now rebuilds the meeting's segments from its
[MM:SS] [speaker] lines and trims the rest; the line parser moves to
@d3ro/core/meeting-transcript and the meeting view uses it too.

Prompt edits of the four desktop presets that exist on the phone update the
server preset row (a reset restores its default; {{targetLanguage}} is sent
as English, the only target on both sides), and edits made on another desktop
come back. The free-prompt preset has no phone counterpart and stays local.

apps/desktop/tests/integration/cross-device-sync.supabase.test.ts

@@ -332,6 +332,40 @@ describe.skipIf(!enabled)('cross-device sync against local Supabase', () => {
     vi.restoreAllMocks()
   }, 60_000)
 
+  it('데스크톱 전사가 폰이 그리는 구간으로 올라가고, 고친 프리셋 프롬프트가 폰 명령에 쓰인다', async () => {
+    const db = testDb.db
+    const meetingId = crypto.randomUUID()
+    const now = Date.now()
+    db.insert(meetingSessions).values({
+      id: meetingId,
+      title: 'segments',
+      status: 'completed',
+      startedAt: now,
+      rawTranscript: '[00:00] one\n[00:04] two\n[00:09] three',
+      createdAt: now,
+      updatedAt: now,
+    }).run()
+    enqueueChange('meetings', meetingId, 'upsert')
+    getCustomInstructionService().update('builtin-summarize', { prompt: '두 줄로 요약' })
+    enqueueChange('builtin_instructions', 'builtin-summarize', 'upsert')
+    const first = await engine.flush()
+    expect(first.errors).toEqual([])
+    const segments = must(await mobile.from('transcripts').select('segment_index,text,edited').eq('meeting_id', meetingId).order('segment_index'))
+    expect(segments.map((s) => s.text)).toEqual(['one', 'two', 'three'])
+
+    db.update(meetingSessions).set({ editedTranscript: '[00:00] [화자 1] one fixed', updatedAt: Date.now() }).where(eq(meetingSessions.id, meetingId)).run()
+    enqueueChange('meetings', meetingId, 'upsert')
+    expect((await engine.flush()).errors).toEqual([])
+    const edited = must(await mobile.from('transcripts').select('text,speaker,edited').eq('meeting_id', meetingId))
+    expect(edited).toEqual([{ text: 'one fixed', speaker: '화자 1', edited: true }])
+
+    const summarize = must(await mobile.from('custom_instructions').select('prompt').eq('user_id', userId).eq('builtin_key', 'summarize').single())
+    expect(summarize.prompt).toBe('두 줄로 요약')
+    // 폰은 여전히 프리셋 행을 직접 고칠 수 없다(RLS)
+    const blocked = await mobile.from('custom_instructions').update({ prompt: 'hack' }).eq('user_id', userId).eq('builtin_key', 'summarize').select('id')
+    expect(blocked.data ?? []).toEqual([])
+  })
+
   it('데스크톱이 기기 목록에 나타나고, 모바일에서 해제하면 revoked가 된다', async () => {
     const info = { deviceName: 'IT-DESKTOP', appVersion: '9.9.9', osVersion: 'test' }
     const first = await checkInDesktopDevice(desktop, userId, 'signin', info, null)