docs: fix invalid nginx error_page 401 example for API routes

The API route example used `error_page 401 =401;`, but nginx parses the
`=401` as the error_page URI, so an unauthenticated API call is internally
redirected to a location named /=401 instead of returning 401 to the client.
Drop the directive: without an error_page for 401, nginx returns the 401 from
auth_request to the client unchanged.

Also add a note that the browser and API snippets show only the routing
difference and the rest of the auth_request config (the auth_request_set /
proxy_set_header lines that pass X-User, X-Email, token and cookies) must be
replicated from the main example.

Fixes #3467.

Signed-off-by: Mateen Anjum <mateenali66@gmail.com>

CHANGELOG.md

@@ -7,6 +7,7 @@
 ## Breaking Changes
 
 ## Changes since v7.15.3
+- [#3469](https://github.com/oauth2-proxy/oauth2-proxy/pull/3469) docs: fix invalid `error_page 401 =401` in the nginx API route example and note that the shared `auth_request` config must be replicated
 
 # V7.15.3
 

docs/docs/configuration/integrations/nginx.md

@@ -119,6 +119,10 @@ The named location pattern above ensures the browser receives a standard **302 r
 Redirecting authentication failures (302 to `/oauth2/sign_in`) should **only be used for browser-facing routes**. API or machine clients should receive a plain 401/403 response without redirect.
 :::
 
+:::note
+The two snippets below show only the routing difference (redirect vs. no redirect). Replicate the rest of the `auth_request` configuration from the [main `location /` example](#configuring-for-use-with-the-nginx-auth_request-directive) above (the `auth_request_set`/`proxy_set_header` lines that pass `X-User`, `X-Email`, the access token and cookies to the backend); copying only the lines shown here omits those headers.
+:::
+
 #### Browser-facing routes (HTML, UI)
 
 For interactive browser routes where users should be redirected to sign in:
@@ -142,7 +146,8 @@ For API endpoints where clients expect a 401/403 status code (not a redirect):
 ```nginx
 location /api/ {
   auth_request /oauth2/auth;
-  error_page 401 =401;  # Pass through the 401 status
+  # No `error_page 401` directive: nginx returns the 401 from auth_request
+  # to the client unchanged, which is what API/machine clients expect.
   proxy_pass http://backend/;
 }
 ```