Replace Date.now() with performance.now() for duration measurements (#258)

* Replace Date.now() with performance.now() for duration measurements

- Import performance from node:perf_hooks in all affected files
- Update Timer and ThinkingTimer classes to use performance.now()
- Update activity tracking (markActivity, getIdleMs) to use performance.now()
- Update cache duration measurements to use performance.now()
- Update agent execution timing (cursor, opencode) to use performance.now()
- Update subprocess execution timing to use performance.now()
- Update API performance monitoring to use performance.now()
- Update prep phase timing to use performance.now()
- Update timer.test.ts to mock performance.now() instead of Date.now()

Benefits:
- Monotonic clock immune to system clock adjustments
- Higher precision (microsecond vs millisecond resolution)
- Purpose-built for performance measurement

Fixes #245

* fix lint.

* Round float durations to integers in logging

Preserve original behavior by rounding performance.now() float values
to integers when displaying/logging millisecond durations.

* fix lint.

---------

Co-authored-by: pullfrog[bot] <226033991+pullfrog[bot]@users.noreply.github.com>
Co-authored-by: Robin Tail <robin_tail@me.com>

prep/index.ts

@@ -1,3 +1,4 @@
+import { performance } from "node:perf_hooks";
 import { log } from "../utils/cli.ts";
 import { installNodeDependencies } from "./installNodeDependencies.ts";
 import { installPythonDependencies } from "./installPythonDependencies.ts";
@@ -14,7 +15,7 @@ const prepSteps: PrepDefinition[] = [installNodeDependencies, installPythonDepen
  */
 export async function runPrepPhase(options: PrepOptions): Promise<PrepResult[]> {
   log.debug("» starting prep phase...");
-  const startTime = Date.now();
+  const startTime = performance.now();
   const results: PrepResult[] = [];
 
   for (const step of prepSteps) {
@@ -35,8 +36,8 @@ export async function runPrepPhase(options: PrepOptions): Promise<PrepResult[]>
     }
   }
 
-  const totalDurationMs = Date.now() - startTime;
-  log.debug(`» prep phase completed (${totalDurationMs}ms)`);
+  const totalDurationMs = performance.now() - startTime;
+  log.debug(`» prep phase completed (${Math.round(totalDurationMs)}ms)`);
 
   return results;
 }