Sync active timers with server, add context filtering, and improve timer formatting

[Fingolfin7]

Motivation

• Ensure the client-side active timers list stays in sync with server state so timers started/stopped outside the UI are reflected quickly.
• Apply the same context-based filtering to active-session listing as is used for other session queries.
• Provide clearer, human-readable duration strings that include days/hours/minutes/seconds and limit DOM updates to active timer cards only.

Description

• Apply filter_by_active_context to the list_active_sessions API endpoint so GET /api/list_active_sessions/ respects the context query parameter.
• Restrict DOM updates in dynamic_timers.js to active timer cards via a focused selector and replace the naive formatter with formatTime that outputs days/hours/minutes/seconds.
• Add getLocalTimerIds and syncActiveTimers in dynamic_timers.js which poll GET /api/list_active_sessions/ and conditionally call window.location.reload() only when the local timer set differs from the server, with awareness of partial lists via data-partial-list and data-max-visible.
• Add `data-partial-list=

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Codex Task