refactor: extract AI checker base class + fix persistence + dedupe restoration

- Add AiCheckerBase abstract class to eliminate ~700 lines of duplication
  between AiIdleChecker and AiPlanChecker
- Fix missing persistence fields (completionConfirmMs, noOutputTimeoutMs)
  that caused custom timing values to be lost on server restart
- Extract restoreRespawnController() helper to dedupe session restoration
  logic between state.json and screens.json paths

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

src/types.ts

@@ -558,6 +558,10 @@ export interface PersistedRespawnConfig {
   autoAcceptPrompts?: boolean;
   /** Delay before auto-accepting prompts (ms) */
   autoAcceptDelayMs?: number;
+  /** Time to wait after completion message before confirming idle (ms) */
+  completionConfirmMs?: number;
+  /** Fallback timeout when no output received at all (ms) */
+  noOutputTimeoutMs?: number;
   /** Whether AI idle check is enabled */
   aiIdleCheckEnabled?: boolean;
   /** Model to use for AI idle check */