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.

ui/src/lib/mock/data.ts

@@ -47,6 +47,9 @@ export const PROFILE: Profile = {
   dictVersions: { erudit_ru: 'v1.3.0', scrabble_ru: 'v1.3.0', scrabble_en: 'v1.3.0' },
   // Post-move interstitial config (short cooldowns so the mock stub is exercisable); not suppressed.
   ads: { cooldownGlobalS: 300, cooldownVsAiS: 1800, cooldownHintS: 60, suppressed: false },
+  // The durable tier's per-kind active-game caps (the seeded defaults); the e2e lowers them through
+  // window.__mock.setGameLimits to exercise the New-Game lock.
+  gameLimits: { vsAi: 10, random: 10, friends: 10 },
 };
 
 // Seed social/account data for the mock (pnpm start + Playwright). The mock profile
@@ -189,6 +192,7 @@ function activeGame(): MockGame {
       vsAi: false,
       unreadChat: false,
       unreadMessages: false,
+      kind: 2,
       status: 'active',
       players: 2,
       toMove: 0,
@@ -227,6 +231,7 @@ function finishedG2(): MockGame {
       vsAi: false,
       unreadChat: false,
       unreadMessages: false,
+      kind: 2,
       status: 'finished',
       players: 2,
       toMove: 0,
@@ -266,6 +271,7 @@ function finishedG3(): MockGame {
       vsAi: false,
       unreadChat: false,
       unreadMessages: false,
+      kind: 2,
       status: 'finished',
       players: 2,
       toMove: 0,