feat(ui): per-kind active-game limit lock on the New Game screen

Carry the caller's per-tier active-game caps and each game's kind on the
wire (Profile.game_limits + GameView.kind, additive FBS + gateway transcode
+ client codec, committed regen). The New Game screen counts the player's
active games per kind from the lobby and locks a capped start: an outline
button with a lock that opens a funnel modal instead of a game -- a sign-in
prompt for a guest, a "finish a current game first" notice for a signed-in
account (native Telegram popup, in-app modal elsewhere). The lock lifts via
the existing profile refetch after a guest->durable upgrade.

Remove the lobby's old at_game_limit New-Game tab disable + notice: the flag
(now the random-kind cap) conflicted with the per-kind lock -- it hid the
screen where the lock lives and wrongly blocked an unfulfilled kind. The New
Game tab is always enabled; the per-kind start lock is the only gate. The
at_game_limit wire field stays (unused by the client) for a later cleanup.

Tests: client lock logic + codec kind/game_limits roundtrip + gateway
transcode encode + native popup builders (unit); a mock e2e for the lock
badge and the modal.

docs/FUNCTIONAL.md

@@ -190,14 +190,17 @@ Mini App, the link only opens the app and the recipient enters the copied code b
 settings and the game starts once every invitee has accepted — any decline cancels it, and an unanswered invitation
 expires after seven days.
 
-**Simultaneous-game cap.** A player may hold at most **10 active quick games** at once —
-counting both in-progress and still-searching auto-match/AI games, but **not** friend games
-created by invitation. At the cap the **New Game** button is disabled (greyed) and a plain,
-low-emphasis line under the lists reads *"You've reached the simultaneous games limit."*; both
-clear automatically once an active game finishes and the count drops below ten. The cap blocks
-**starting** any game (a quick game or a friend invitation, which share the New Game entry), but
-**accepting an incoming invitation is never blocked** — so friend games are capped from the other
-end (you cannot initiate one while at the cap) while you can always join a game you are invited to.
+**Active-game caps (per kind).** A player's simultaneous unfinished games are capped **per kind** —
+against the AI, in a random match, and by friend invitation — with separate limits for a **guest**
+and a signed-in (durable) account, tuned by the operator without a release. A **guest** may hold **1**
+game against the AI and **1** random match at once and cannot start friend games at all; a signed-in
+account holds up to **10** of each kind. Only in-progress and still-searching games count; a finished
+game frees its slot, and games already in progress are never interrupted. On the **New Game** screen a
+start whose kind is at its cap shows a **🔒** and, when tapped, opens a short prompt instead of a game:
+a **guest** is invited to sign in or create an account (to play several games at once), a signed-in
+player is told to finish a current game first. **Accepting an incoming invitation is never blocked** —
+friend games are capped only when you start one. The server enforces every cap regardless of the
+client, and refuses a guest's friend request, friend code or invitation outright.
 
 ### Playing a game
 Place tiles, pass, exchange, or resign. Pass and exchange share one control —