Fixes pingdotgg#978
Stacked on top of pingdotgg#1133 (see https://pr-navigator.pages.dev/t3code-utkarsh).

What Changed

This extends plan consumption to the Implement in new thread flow (pingdotgg#1133).

PR 1 introduced the shared plan-consumption plumbing and applied it to same-thread Implement. This PR wires the remaining caller: when the user chooses Implement in new thread, the new thread’s thread.turn.start command now includes sourceProposedPlan.

That allows the existing orchestration and runtime-ingestion path to mark the source proposed plan as implemented when the new thread’s implementation turn actually starts, so the original plan thread no longer remains actionable.

Why

Before this change, the same-thread implementation flow consumed the plan, but the new-thread implementation flow still started an implementation turn without identifying which source plan it came from.

That left the original thread in a stale actionable state even though the plan was already being implemented elsewhere.

This PR keeps the behavior consistent across both implementation entrypoints by reusing the same source-plan mechanism introduced in PR 1.

UI Changes

Combined the before/after videos and posted it on X because the files' sizes were too large to upload here: https://x.com/UtkarshUsername/status/2033161459447087223?s=20

Stack Context

This PR is stacked on top of:

• fix/consume-plan-on-same-thread-implementation

After PR 1 merges, this PR will be retargeted to main.

Checklist

• This PR is small and focused
• I explained what changed and why
• I included a video for animation/interaction changes