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/helpers/fakeSyncRemote.ts

@@ -156,6 +156,36 @@ export class FakeSyncRemote implements SyncRemote {
     for (let i = rows.length - 1; i >= 0; i--) if (rows[i][parentColumn] === parentId) rows.splice(i, 1)
   }
 
+  async deleteChildrenFrom(
+    table: string,
+    parentColumn: string,
+    parentId: string,
+    indexColumn: string,
+    fromIndex: number
+  ): Promise<void> {
+    this.guard()
+    const rows = this.rows(table)
+    for (let i = rows.length - 1; i >= 0; i--) {
+      if (rows[i][parentColumn] === parentId && Number(rows[i][indexColumn]) >= fromIndex) rows.splice(i, 1)
+    }
+  }
+
+  /** 서버 프리셋 기본 프롬프트 (마이그레이션 0036 과 같은 문구) */
+  static readonly BUILTIN_DEFAULTS: Record<string, string> = {
+    translate_en: 'Translate the following text into natural English. Return only the translation.',
+    summarize: 'Summarize the following text in no more than three concise lines.',
+    formal: 'Rewrite the following text in a formal business style while preserving its meaning. Return only the rewritten text.',
+    explain_code: 'Explain the following code in Korean, including its responsibilities and important caveats.',
+  }
+
+  private ensureBuiltins(): void {
+    for (const [key, prompt] of Object.entries(FakeSyncRemote.BUILTIN_DEFAULTS)) {
+      if (!this.rows('custom_instructions').some((r) => r.builtin_key === key)) {
+        this.mobileInsert('custom_instructions', { id: crypto.randomUUID(), builtin_key: key, name: key, prompt, icon: 'x', sort_order: 10 })
+      }
+    }
+  }
+
   async invokeFunction(name: string, body: Record<string, unknown>): Promise<unknown> {
     this.guard()
     this.invoked.push(`${name}:${String(body.document_id ?? '')}`)
@@ -298,6 +328,26 @@ export class FakeSyncRemote implements SyncRemote {
       else this.mobileInsert('user_settings', { active_instruction_id: instructionId, revision: 1 })
       return row
     }
+    if (name === 'sync_set_builtin_instruction_prompt_v1') {
+      this.ensureBuiltins()
+      const key = String(params.p_builtin_key)
+      const fallback = FakeSyncRemote.BUILTIN_DEFAULTS[key]
+      if (!fallback) throw new SyncRemoteError('invalid_builtin_key', '22023', false)
+      const next = typeof params.p_prompt === 'string' && params.p_prompt.trim() ? params.p_prompt.trim() : fallback
+      const row = this.rows('custom_instructions').find((r) => r.builtin_key === key)!
+      if (row.prompt !== next) Object.assign(row, { prompt: next, revision: Number(row.revision) + 1, updated_at: this.now() })
+      return row
+    }
+    if (name === 'sync_list_builtin_instructions_v1') {
+      return this.rows('custom_instructions')
+        .filter((r) => typeof r.builtin_key === 'string')
+        .map((r) => ({
+          builtin_key: r.builtin_key,
+          prompt: r.prompt,
+          is_default: r.prompt === FakeSyncRemote.BUILTIN_DEFAULTS[String(r.builtin_key)],
+          updated_at: r.updated_at,
+        }))
+    }
     if (name === 'sync_delete_user_template_v1') {
       const id = String(params.p_id)
       if (!this.find('user_templates', id)) return false