Offer a Continue link for ongoing games in the lobby

A player who accidentally closes the game page can now get back to it
from the lobby.

Backend:
- GameStore.list_ongoing() returns live unfinished sessions (newest
  first), skipping expired keys and finished matches;
- GET /api/me/games lists the games the caller is seated in, with the
  game name/description, players, open seats and your seat (no join
  code);
- store, platform route and app integration tests.

Frontend:
- MyGame/MyGamesPage model plus api::my_games();
- the lobby shows a "Your games" card with phase, seats, players and a
  Continue button linking to /game/<id>; hidden when empty or logged
  out.

web/src/model.rs

@@ -498,6 +498,66 @@ pub struct PreviewPlayer {
     pub team: Option<String>,
 }
 
+/// One ongoing game the caller is seated in (GET /api/me/games): the
+/// lobby payload plus the game's display name, for the "Continue" list.
+#[derive(Debug, Clone, Deserialize)]
+#[allow(dead_code)]
+pub struct MyGame {
+    pub id: String,
+    #[serde(default)]
+    pub game_type: String,
+    #[serde(default)]
+    pub game_name: String,
+    #[serde(default)]
+    pub game_description: String,
+    #[serde(default)]
+    pub phase: String,
+    #[serde(default)]
+    pub players: Vec<PreviewPlayer>,
+    #[serde(default)]
+    pub seats_open: Option<usize>,
+    #[serde(default)]
+    pub your_seat: Option<usize>,
+    #[serde(default)]
+    pub created_at: Option<String>,
+}
+
+impl MyGame {
+    /// Human-readable phase for the lobby list.
+    pub fn phase_label(&self) -> &'static str {
+        match self.phase.as_str() {
+            "lobby" => "Waiting for players",
+            "hand_end" => "Between hands",
+            _ => "In progress",
+        }
+    }
+
+    /// Seated/total note, e.g. `"2/4 seated"`.
+    pub fn seats_text(&self) -> String {
+        match self.seats_open {
+            Some(open) => {
+                format!("{}/{} seated", self.players.len(), self.players.len() + open)
+            }
+            None => format!("{} seated", self.players.len()),
+        }
+    }
+
+    /// The seated players' names, for the list subtitle.
+    pub fn players_text(&self) -> String {
+        self.players
+            .iter()
+            .map(|p| p.name.clone())
+            .collect::<Vec<_>>()
+            .join(", ")
+    }
+}
+
+#[derive(Debug, Clone, Deserialize)]
+pub struct MyGamesPage {
+    #[serde(default)]
+    pub results: Vec<MyGame>,
+}
+
 /// One option to show in the join preview, rendered from the game's
 /// schema and the engine-echoed value.
 #[derive(Debug, Clone, PartialEq)]
@@ -1218,6 +1278,41 @@ mod tests {
         assert_eq!("C", g.player_name_for_team("C"));
     }
 
+    #[test]
+    fn my_game_phase_and_seats_text() {
+        let g: MyGame = serde_json::from_value(json!({
+            "id": "g1",
+            "game_type": "briscola_a_quattro",
+            "game_name": "Briscola a quattro",
+            "phase": "lobby",
+            "players": [
+                {"name": "Alice", "seat": 0, "team": "A"},
+                {"name": "Bob", "seat": 1, "team": "B"},
+            ],
+            "seats_open": 2,
+        }))
+        .unwrap();
+        assert_eq!("Waiting for players", g.phase_label());
+        assert_eq!("2/4 seated", g.seats_text());
+        assert_eq!("Alice, Bob", g.players_text());
+
+        let playing: MyGame = serde_json::from_value(json!({
+            "id": "g2",
+            "game_type": "briscola",
+            "phase": "playing",
+        }))
+        .unwrap();
+        assert_eq!("In progress", playing.phase_label());
+        assert_eq!("0 seated", playing.seats_text());
+
+        let between: MyGame = serde_json::from_value(json!({
+            "id": "g3",
+            "phase": "hand_end",
+        }))
+        .unwrap();
+        assert_eq!("Between hands", between.phase_label());
+    }
+
     #[test]
     fn fallback_game_types_cover_the_registered_games() {
         let types = fallback_game_types();