Components.AI: Preserve composer drafts across transitions and conversation switches

### Is your feature request related to a problem? Please describe the problem.

Typed input can be lost across render-mode transitions or when moving between conversations.

### Describe the solution you'd like

Define and implement draft identity and lifetime so each conversation restores its text and applicable composer state across switching and render-mode transitions without silently losing user input.

### Additional context

Leaf task for #68227 requirement C7. Relates to conversation-switch semantics.

Start by reading requirement C7 in issue #68227 and the related conversation-switch semantics. Locate the Components.AI composer entry point, then define how draft identity and lifetime should work across conversation switches and render-mode transitions; done means text and applicable composer state are restored without silent loss.

Written by the indexing model from the issue text.