Follow-ups of agent mode selection

• If you change the default agent, the mode selector only updates after the first key stroke
• Mode selection doesn't persist when switching between chat sessions (resets to default when you come back to a chat, which might not be the one you used in the previous request)
• A keyboard shortcut or adding it to the tab order for cycling modes via the keyboard would be nice (since the chat-input:cycle-mod command exists but isn't used yet) Edit ndoschek: This already works fine via Shit+Tab 👍
• Add the mode selector also to the Ask AI input (Ctrl+I) overlay in the editor.
• If agent is selected/pinned/default in the chat input and you change the default prompt (mode) in the AI Configuration, the mode is not updated correctly (e.g., for Coder) (fixed via #17271)

Originally posted by @ndoschek in https://github.com/eclipse-theia/theia/pull/16860#pullrequestreview-3715381447