feat: sparser robot nudges, typed unread badge, lobby unread bump

Three owner-requested polish changes:

- robot: replace the lengthening 60-90 min -> 6 h proactive-nudge ramp with a
  flat uniform 9-12 h wait before every nudge; the existing sleep-window gate
  still skips and defers a nudge that would land in the robot's night.
- ui: colour the lobby/in-game unread dot by type -- the regular danger colour
  when a chat message is unread, a softer amber (--warn) when only nudges are.
  Adds a per-viewer unread_messages flag (chat_messages.kind='message') across
  the backend DTO, FlatBuffers wire, gateway transcode and the UI store.
- ui: float games with any unread notification to the top of the lobby's
  your-turn and opponent-turn sections (finished keeps its order), reusing the
  existing unread_chat flag.

Docs (ARCHITECTURE 7, FUNCTIONAL + _ru) updated. No DB migration; the new wire
field is backward-compatible.

backend/internal/server/dto.go

@@ -112,6 +112,10 @@ type gameDTO struct {
 	// entry (message or nudge) in this game. It seeds the lobby and in-game unread badge.
 	// gameDTOFromGame leaves it false (it has no viewer); the handlers fill it.
 	UnreadChat bool `json:"unread_chat"`
+	// UnreadMessages is a per-viewer flag: at least one of the unread entries is a real chat
+	// message (not just a nudge). It colours the badge — set with UnreadChat it is the regular
+	// badge, UnreadChat alone is the softer nudge-only badge. gameDTOFromGame leaves it false.
+	UnreadMessages bool `json:"unread_messages"`
 }
 
 // moveResultDTO is the outcome of a committed move. Rack carries the actor's refilled rack as