docs: stop presenting the in-memory client as the way to connect (#3443)

docs/run/asgi.md

@@ -137,4 +137,4 @@ A browser-based client needs two permissions from you: to **send** its MCP reque
 * Browser clients need CORS: `allow_headers` for the `Mcp-*` request headers, `expose_headers=["Mcp-Session-Id"]` for the response.
 * `@mcp.custom_route()` adds plain, unauthenticated HTTP endpoints next to `/mcp`.
 
-Once the server is reachable at a real URL, **[The Client](../client/index.md)** connects to it with that URL instead of a server object.
+Once the server is reachable at a real URL, **[The Client](../client/index.md)** connects to it with that URL.

docs/run/authorization.md

@@ -124,6 +124,6 @@ An authorization server can also accept an enterprise identity provider's signed
 * `token_verifier=` and `auth=AuthSettings(issuer_url=..., resource_server_url=..., required_scopes=[...])` always travel together.
 * The SDK publishes [RFC 9728](https://datatracker.ietf.org/doc/html/rfc9728) Protected Resource Metadata at `/.well-known/oauth-protected-resource/...` and answers unauthenticated requests with a 401 whose `WWW-Authenticate` header points at it. That is the entire discovery story.
 * `get_access_token()` in any handler is who's calling.
-* Authorization is an HTTP concern. `stdio` and the in-memory client never see it.
+* Authorization is an HTTP concern. `stdio` and the in-memory test client never see it.
 
 The client half (discovering your authorization server and fetching the token for you) is **[OAuth clients](../client/oauth-clients.md)**. And a client that *asserts* an identity instead of asking a user for one is **[Identity assertion](../client/identity-assertion.md)**.

docs/run/legacy-clients.md

@@ -16,15 +16,25 @@ So a legacy client is not something you build *for*. It is something that connec
 
 ## One handler, both eras
 
-Here is a tool that has to ask the user something, and both eras of client calling it:
+Here is a tool that has to ask the user something:
 
-```python title="server.py" hl_lines="24 37-38"
+```python title="server.py" hl_lines="21"
 --8<-- "docs_src/legacy_clients/tutorial001.py"
 ```
 
 `reserve` needs one thing the model didn't supply: how many copies. `Annotated[..., Resolve(ask_quantity)]` is how a tool declares that (**[Dependencies](../handlers/dependencies.md)** is that whole story). Nothing in `reserve` names a version, checks a capability, or branches.
 
-The two clients are open **at the same time**, on the same `mcp` object. `mode="legacy"` runs the `initialize` handshake: the exact connection a pre-2026 client opens. The other one takes the default and lands on `2026-07-28`.
+Serve it over HTTP, and here are both eras of client calling it:
+
+```console
+uv run mcp run server.py --transport streamable-http
+```
+
+```python title="client.py" hl_lines="14-15"
+--8<-- "docs_src/legacy_clients/tutorial001_client.py"
+```
+
+The two clients are open **at the same time**, against the same running server. `mode="legacy"` runs the `initialize` handshake: the exact connection a pre-2026 client opens. The other one takes the default and lands on `2026-07-28`. Run `python client.py` from a second terminal:
 
 ```text
 2025-11-25 {'result': "Reserved 2 of 'Dune'."}
@@ -115,7 +125,7 @@ Two things about it matter more than what it does.
 
 !!! check
     Do the wrong thing. `reserve` is the exact tool that just served both clients. Deploy it with
-    `stateless_http=True`, connect the same two clients over HTTP, and call it from each.
+    `stateless_http=True`, connect the same two clients, and call it from each.
 
     The modern client still gets `Reserved 2 of 'Dune'.` The modern leg didn't change.