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 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. That prompt, including the fixed text and every label, stays within PROVIDER_SEND_TURN_MAX_INPUT_CHARS. When continuation is off, or it fails, only the still-open tasks are settled once as stopped and the thread gets the existing restart notice.

An empty open-task list writes continueAfterServerUpdateTasks: null, so a stale recorded list is not reused.

A retry from a persisted starting session still restores recorded task names whenever the continuation marker is present, including after settlement already wrote task.completed rows. Promptless providers receive those names instead of { continuation: true }. listUnterminatedTasks skips activity rows whose payload is not valid JSON, and plan or dream tasks stay excluded when any decisive row has that 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.

This redo also closes the review findings from pingdotgg#13621: an oversized task label could make sendTurn reject the continuation, a stale task marker could settle the same tasks again, and settlement could dispatch a second completion for tasks that were already recorded.

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