fix(server): continue threads whose background tasks outlived the turn

[macodev00]

What Changed

Restart continuation treats open background tasks the same as an interrupted turn.

markRunningProviderSessionsForContinuation and reconcileProviderSessions now notice unterminated task.* rows (the persisted form of Claude liveTaskIds). When "Continue threads after restarts" is on, the thread is continued with a prompt that names the stopped tasks. When it is off, or continuation fails, those tasks are settled as stopped and the thread gets the existing restart notice.

This also covers the three restart-recovery cases that blocked the previous attempt:

1. A retry from a persisted starting session restores recorded task names whenever the continuation marker is present, including after settleStoppedTasks has already written task.completed rows. Promptless providers still receive the named tasks instead of { continuation: true }.
2. listUnterminatedTasks skips activity rows whose payload_json is not valid JSON, so one malformed payload cannot abort continuation for every thread.
3. Plan and dream tasks stay excluded when any decisive row for that task has taskType plan or dream, even if a later status update omits taskType.

Why

A Claude thread whose turn has already finished can still be running subagents or background shells. The session is ready with no active turn, so a restart neither continued it nor told the user the work was lost.

Fixes pingdotgg#13400

UI Changes

None.

Checklist

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