Address Greptile 4/5 feedback: auto-clear idle badge, restore status color, fix display gap

- Clear idle_warning errorCode in onLog when output resumes, so the
  orange Idle badge disappears without requiring an API call
- Restore idle_timeout entry in status-colors.ts for errorCode-based
  badge styling
- Simplify LiveRunWidget time display to a single if/else chain,
  eliminating the gap for non-active runs with lastOutputAt but no
  finishedAt

Co-Authored-By: Paperclip <noreply@paperclip.ing>

server/src/services/heartbeat.ts

@@ -2517,6 +2517,10 @@ export function heartbeatService(db: Db) {
           await db.update(heartbeatRuns)
             .set({ lastOutputAt: lastOutputAtLatest, updatedAt: new Date() })
             .where(eq(heartbeatRuns.id, runId));
+          // Clear idle_warning if set — output resumed
+          await db.update(heartbeatRuns)
+            .set({ errorCode: null, error: null })
+            .where(and(eq(heartbeatRuns.id, runId), eq(heartbeatRuns.errorCode, IDLE_WARNING_ERROR_CODE)));
           lastOutputAtTimer = setTimeout(() => {
             lastOutputAtFlushPending = false;
             lastOutputAtTimer = null;

ui/src/components/LiveRunWidget.tsx

@@ -125,18 +125,13 @@ export function LiveRunWidget({ issueId, companyId }: LiveRunWidgetProps) {
                         Idle
                       </span>
                     )}
-                    {run.lastOutputAt && isRunActive(run.status) && (
+                    {isRunActive(run.status) && run.lastOutputAt ? (
                       <span title={`Last output: ${formatDateTime(run.lastOutputAt)}`}>
                         Last output {relativeTime(run.lastOutputAt)}
                       </span>
-                    )}
-                    {run.finishedAt && !isRunActive(run.status) && (
+                    ) : run.finishedAt ? (
                       <span>{formatDateTime(run.finishedAt)}</span>
-                    )}
-                    {!run.lastOutputAt && isRunActive(run.status) && (
-                      <span>{formatDateTime(run.startedAt ?? run.createdAt)}</span>
-                    )}
-                    {!run.finishedAt && !isRunActive(run.status) && !run.lastOutputAt && (
+                    ) : (
                       <span>{formatDateTime(run.startedAt ?? run.createdAt)}</span>
                     )}
                   </div>

ui/src/lib/status-colors.ts

@@ -55,6 +55,7 @@ export const statusBadge: Record<string, string> = {
   // Run statuses
   failed: "bg-red-100 text-red-700 dark:bg-red-900/50 dark:text-red-300",
   timed_out: "bg-orange-100 text-orange-700 dark:bg-orange-900/50 dark:text-orange-300",
+  idle_timeout: "bg-orange-100 text-orange-700 dark:bg-orange-900/50 dark:text-orange-300",
   succeeded: "bg-green-100 text-green-700 dark:bg-green-900/50 dark:text-green-300",
   error: "bg-red-100 text-red-700 dark:bg-red-900/50 dark:text-red-300",
   terminated: "bg-red-100 text-red-700 dark:bg-red-900/50 dark:text-red-300",