Add room_types and room_type to /publicRooms (#1199)

MSC: https://github.com/matrix-org/matrix-spec-proposals/pull/3827
Incorporates https://github.com/matrix-org/matrix-spec-proposals/pull/3858

data/api/client-server/definitions/public_rooms_chunk.yaml

@@ -57,6 +57,11 @@ properties:
       The room's join rule. When not present, the room is assumed to
       be `public`.
     example: "public"
+  room_type:
+    type: string
+    x-addedInMatrixVersion: "1.4"
+    description: |-
+      The `type` of room (from [`m.room.create`](/client-server-api/#mroomcreate)), if any.
 required:
   - room_id
   - num_joined_members

data/api/client-server/definitions/public_rooms_response.yaml

@@ -62,7 +62,8 @@ example: {
       "room_id": "!ol19s:bleecker.street",
       "topic": "Tasty tasty cheese",
       "world_readable": true,
-      "join_rule": "public"
+      "join_rule": "public",
+      "room_type": "m.space"
     }
   ],
   "next_batch": "p190q",