refactor(prompt-callers): migrate shared and cli dispatch

Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)

Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>

src/cli/run/runner.ts

@@ -13,7 +13,7 @@ import { loadAgentProfileColors } from "./agent-profile-colors"
 import { suppressRunInput } from "./stdin-suppression"
 import { createTimestampedStdoutController } from "./timestamp-output"
 import { createCliPostHog, getPostHogDistinctId } from "../../shared/posthog"
-import { promptAsyncAfterSessionIdle } from "../../shared/prompt-async-gate"
+import { dispatchInternalPrompt } from "../../shared/prompt-async-gate"
 
 export { resolveRunAgent }
 
@@ -110,7 +110,8 @@ export async function run(options: RunOptions): Promise<number> {
         () => {},
       )
 
-      const promptResult = await promptAsyncAfterSessionIdle({
+      const promptResult = await dispatchInternalPrompt({
+        mode: "async",
         client,
         sessionID,
         source: "cli-run",