[Feature Request] Granular per-category notification preferences (desktop/mobile) + don't suppress blocking prompts on mobile when terminal is focused

Summary

Claude Code currently suppresses both desktop and mobile notifications while the terminal has OS focus, with no way to opt out, and the set of notifications can't be tuned per type. For users who operate Claude Code remotely from the phone, this makes away-from-desk work unreliable. The ideal fix is granular, per-category notification preferences — each user picks which events reach the phone.

The real use case

The key scenario isn't "at my desk in another window" — it's operating Claude Code remotely from the phone (via Remote Control) while away from the machine. In that mode the notifications that matter most are the interactive / blocking ones:

• a question / multiple-choice selection is waiting,
• a permission / approval prompt is blocking,
• the agent is idle, waiting for input.

If these don't reliably reach the phone, the session silently stalls with no way to know from outside — the task is stuck until I walk back to the terminal.

Why granular settings (not "always notify")

Always-on notifications annoy many users, so a blanket "notify everything" isn't the answer. Different people want different events. The strongest solution is letting each user toggle notifications per category and per destination (desktop / mobile) — e.g. someone working remotely turns ON mobile push for approval + selection + input-needed, and leaves routine/completion OFF.

Requested

1. Per-category notification preferences, ideally split by destination (desktop vs mobile). Suggested categories:

• question / multiple-choice selection waiting
• permission / approval prompt blocking
• input-needed / idle waiting
• task completion
• routine progress

1. Allow the interactive/blocking categories to be delivered to mobile independent of terminal focus (these are blocking states, not routine noise). A "remote (away) mode" preset that enables exactly these would be ideal.
2. Document the focus-gating behavior in the notifications docs — it's currently only discoverable via tool return strings (Not sent — terminal has focus... vs Mobile push requested), and docs implying "no presence detection" sent debugging down the wrong path.

Environment

• Claude Code CLI on macOS, Remote Control connected, official iPhone app push.