feat(webui): support caching user input during generation in the frontend

**Background**

Currently, the input button behaves as follows:
- When the agent is idle, the button shows an arrow (submit) icon
- When the agent is actively replying, the button shows a stop icon

However, there is a missing interaction: when the agent is generating a response **and** the user has already typed something in the input box, the button should transition to a **send** state. Upon clicking, the input should be **cached locally on the frontend** and held until the agent finishes its current reply (i.e., upon receiving a `ReplyEndEvent`), at which point the cached input is automatically submitted to the backend — a deferred send. Note that this is purely frontend logic with no backend changes required.

**Changes**
- Update the send button icon rendering logic in the text input component to handle the new intermediate state
- Add a component to display the cached/pending user input in the text input area
- Implement the deferred send logic to automatically flush the cached input once the agent reply ends