[AiAssistance] Refactor ChatView for object parameters in render methods

Drive-by:
* Updated `handle` and `on` prefix usages. (`on` is for inputs
and `handle` is for actual handlers)
* Separated methods and regular properties in object parameters.

Bug: 394005781
Change-Id: Ib3739fae7176f1d98f05f8c800618c484cbb34fe
Reviewed-on: https://chromium-review.googlesource.com/c/devtools/devtools-frontend/+/6252785
Auto-Submit: Ergün Erdoğmuş <[email protected]>
Commit-Queue: Ergün Erdoğmuş <[email protected]>
Reviewed-by: Alex Rudenko <[email protected]>

Author: ergunsh