The cursor in terminal vi mode was not visually updating when using
vi motion navigation keys, despite internal position tracking working correctly.
This was caused by missing `cx.notify()` calls to trigger re-rendering.

Refactored keystroke handling by extracting common logic from `key_down`
and `send_keystroke` into a shared `process_keystroke` helper method.

Closes #46736

Release Notes:

- terminal: Fixed vi mode cursor not visually updating when navigating
with vi motion keys

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Co-authored-by: dino <dinojoaocosta@gmail.com>