Improve active-timers syncing and add home-specific timer script

[Fingolfin7]

Motivation

• Ensure the active-timers UI remains consistent with server state and supports pages that show partial/limited timer lists.
• Reduce unnecessary DOM traversal and improve time formatting to include days/hours for long-running timers.
• Provide a specialized, lightweight client-side refresh for the home page that can reload only the timers fragment when needed.

Description

• Apply filter_by_active_context to list_active_sessions in core/api.py so the active-sessions API respects the same context filtering as other session endpoints.
• Update core/static/core/js/dynamic_timers.js to scope DOM queries to #active-timers .card[id^="timer-"], add formatTime that includes days/hours, implement getLocalTimerIds and syncActiveTimers which polls /api/list_active_sessions/ and reloads the page when the server-side active timers differ from the client.
• Add a new core/static/core/js/home_timers.js that mirrors the duration updates but is home-page aware: it checks capacity via data-max-visible, refreshes the timers fragment via $(selector).load() when needed, and guards concurrent refreshes with refreshInFlight.
• Update core/templates/core/home.html to load home_timers.js instead of dynamic_timers.js and add data-max-visible="5" to #active-timers for partial-list behavior.

Testing

• No automated tests were run as part of this change.

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Codex Task