Fix input persistence when sending to new chat session (#293577)

* Initial plan

* Fix: Clear input when sending to new chat session

When using Ctrl+Shift+Enter to send a prompt to a new chat session,
the original chat's input is now properly cleared by calling
widget.setInput('') before creating the new session. This ensures
that when navigating back to the original session, the input field
is empty as expected.

Co-authored-by: roblourens <323878+roblourens@users.noreply.github.com>

* Add test for SendToNewChatAction input clearing

Co-authored-by: roblourens <323878+roblourens@users.noreply.github.com>

* Improve test to verify operation sequence

Updated test to assert that input clearing happens before session
clearing, ensuring the operations occur in the correct order.

Co-authored-by: roblourens <323878+roblourens@users.noreply.github.com>

* Replace dynamic import with static import

Use static import and registerChatExecuteActions() call instead of
require() to follow VS Code testing patterns.

Co-authored-by: roblourens <323878+roblourens@users.noreply.github.com>

* Remove test rather than fix it

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Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: roblourens <323878+roblourens@users.noreply.github.com>
Co-authored-by: Rob Lourens <roblourens@gmail.com>