refactor: extract standalone chat package from game-specific handlers

Create chat/ package with Message type, Room (NATS pub/sub + buffer),
DB persistence helpers, and a unified templ component parameterized by
Config (CSS prefix, post URL, color function, key propagation).

Both c4game and snakegame now use chat.Room for message management and
chatcomponents.Chat for rendering, eliminating the duplicated
ChatMessage types, chat templ components, chatAutoScroll scripts,
color functions, and inline buffer management.

game/types.go

@@ -69,11 +69,3 @@ func (g *Game) WinningCellsFromJSON(data string) error {
 	}
 	return json.Unmarshal([]byte(data), &g.WinningCells)
 }
-
-// ChatMessage is the domain type for persisted C4 chat messages.
-type ChatMessage struct {
-	Nickname string `json:"nickname"`
-	Color    int    `json:"color"` // 1=Red, 2=Yellow
-	Message  string `json:"message"`
-	Time     int64  `json:"time"`
-}