docs(examples): rename MCP_AUTH_PROTOCOL and update guides

Update multiprotocol docs/READMEs to use MCP_AUTH_PROTOCOL (no PHASE wording) and align example instructions with the hardened scripts.

Author: nypdmax

docs/authorization-multiprotocol.md

@@ -467,7 +467,7 @@ See `RequireAuthMiddleware` and PRM handler in `mcp.server.auth` for how these a
 
 - Starts the multi-protocol resource server (`simple-auth-multiprotocol-rs`) on port 8002 with `--api-keys=demo-api-key-12345`. For OAuth, also starts the AS (`simple-auth-as`) on port 9000.
 - Waits for PRM: `GET http://localhost:8002/.well-known/oauth-protected-resource/mcp`.
-- Runs the client according to `MCP_PHASE2_PROTOCOL`:
+- Runs the client according to `MCP_AUTH_PROTOCOL`:
   - **api_key** (default): `simple-auth-multiprotocol-client` with `MCP_SERVER_URL=http://localhost:8002/mcp` and `MCP_API_KEY=demo-api-key-12345`. No AS required.
   - **oauth**: `simple-auth-client` against the same RS; user completes OAuth in the browser, then runs `list`, `call get_time {}`, `quit`.
   - **mutual_tls**: same multiprotocol client without API key; mTLS is a placeholder (no real client cert).

examples/README.md

@@ -11,7 +11,7 @@ for real-world servers.
 **API Key**
 
 - Use `MCP_API_KEY` on the client; start RS with `--api-keys=...` (no AS required).
-- One-command test (from repo root): `MCP_PHASE2_PROTOCOL=api_key ./scripts/run_phase2_multiprotocol_integration_test.sh`
+- One-command test (from repo root): `MCP_AUTH_PROTOCOL=api_key ./scripts/run_phase2_multiprotocol_integration_test.sh`
 
 **OAuth + DPoP**
 
@@ -20,6 +20,7 @@ for real-world servers.
 
 **Mutual TLS (placeholder)**
 
-- mTLS is a placeholder (no client cert validation). Script: `MCP_PHASE2_PROTOCOL=mutual_tls ./scripts/run_phase2_multiprotocol_integration_test.sh`
+- mTLS is a placeholder (no client cert validation). Script: `MCP_AUTH_PROTOCOL=mutual_tls ./scripts/run_phase2_multiprotocol_integration_test.sh`
+- mTLS is a placeholder (no client cert validation). Script: `MCP_AUTH_PROTOCOL=mutual_tls ./scripts/run_phase2_multiprotocol_integration_test.sh`
 
 **Client**: [simple-auth-multiprotocol-client](clients/simple-auth-multiprotocol-client/) — supports API Key (`MCP_API_KEY`), OAuth+DPoP (`MCP_USE_OAUTH=1`, `MCP_DPOP_ENABLED=1`), and mTLS placeholder.

examples/clients/simple-auth-multiprotocol-client/README.md

@@ -26,7 +26,7 @@ MCP_SERVER_URL=http://localhost:8002/mcp MCP_API_KEY=demo-api-key-12345 uv run m
 ```
 
 **One-command test** from repo root:  
-`MCP_PHASE2_PROTOCOL=api_key ./scripts/run_phase2_multiprotocol_integration_test.sh`  
+`MCP_AUTH_PROTOCOL=api_key ./scripts/run_phase2_multiprotocol_integration_test.sh`  
 starts the resource server and this client with API Key; at `mcp>` run `list`, `call get_time {}`, `quit`.
 
 ## Running with OAuth + DPoP
@@ -51,10 +51,10 @@ Complete OAuth in the browser; then at `mcp>` run `list`, `call get_time {}`, `q
 
 Mutual TLS is a **placeholder** in this example: the client registers the `mutual_tls` protocol but does **not** perform client certificate authentication. Selecting mTLS will show a "not implemented" style message.
 
-- **`MCP_PHASE2_PROTOCOL=mutual_tls`** (with the phase2 script) runs this client in mTLS mode; the client will start but mTLS auth is not implemented.
+- **`MCP_AUTH_PROTOCOL=mutual_tls`** (with the phase2 script) runs this client in mTLS mode; the client will start but mTLS auth is not implemented.
 
 **One-command test** from repo root:  
-`MCP_PHASE2_PROTOCOL=mutual_tls ./scripts/run_phase2_multiprotocol_integration_test.sh`
+`MCP_AUTH_PROTOCOL=mutual_tls ./scripts/run_phase2_multiprotocol_integration_test.sh`
 
 ## Commands
 

examples/servers/simple-auth-multiprotocol/README.md

@@ -36,7 +36,7 @@ You can run the Resource Server **without** the Authorization Server when using
 3. At the `mcp>` prompt, run `list`, `call get_time {}`, then `quit`.
 
 **One-command verification** (from repo root):  
-`MCP_PHASE2_PROTOCOL=api_key ./scripts/run_phase2_multiprotocol_integration_test.sh`  
+`MCP_AUTH_PROTOCOL=api_key ./scripts/run_phase2_multiprotocol_integration_test.sh`  
 This starts the RS, then the client with API Key; complete the session with `list`, `call get_time {}`, `quit`.
 
 ## Running with DPoP (OAuth + DPoP)
@@ -66,7 +66,7 @@ Mutual TLS is a **placeholder** in this example: the server accepts the `mutual_
 
 - **Server**: No extra flags; `auth_protocols` already includes `mutual_tls`.
 - **Client** (from repo root):  
-  `MCP_PHASE2_PROTOCOL=mutual_tls ./scripts/run_phase2_multiprotocol_integration_test.sh`  
+  `MCP_AUTH_PROTOCOL=mutual_tls ./scripts/run_phase2_multiprotocol_integration_test.sh`  
   The client will start but mTLS authentication is not implemented in this example.
 
 ## Options