docs: register MCP servers at user scope, not the default

Claude Code's default MCP scope is `local`, which is per-project: the
server is stored under that directory in ~/.claude.json and does not
appear in `claude mcp list` from anywhere else. docs/LOCAL.md showed the
default and so produced exactly that confusion.

User scope is right here — the point is to ask about coursework from any
directory. Project scope would write a .mcp.json into the repo, which is
wrong when the HTTP config carries a bearer token.

Also adds the symptom and the fix, since "it is not in the list" has one
overwhelmingly likely cause.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

docs/LOCAL.md

@@ -39,25 +39,31 @@ Claude Code supports MCP over stdio, SSE and HTTP. The VS Code extension runs
 the same Claude Code underneath, so it reads the same configuration — register
 once and both work.
 
+**Use `--scope user`.** Claude Code's default scope is `local`, which is
+*per-project*: the server is stored under that directory in `~/.claude.json` and
+`claude mcp list` will not show it from anywhere else. Since the point is to ask
+about your coursework from wherever you happen to be, user scope is what you
+want. (`project` scope would write a `.mcp.json` into the repo — wrong here,
+because the HTTP config carries a token.)
+
 **HTTP — matches the deployed setup:**
 
 ```bash
-claude mcp add --transport http schulcloud http://127.0.0.1:8080/mcp \
+claude mcp add --scope user --transport http schulcloud http://127.0.0.1:8080/mcp \
   --header "Authorization: Bearer $MCP_AUTH_TOKEN"
 ```
 
 **stdio — no server or Docker needed:**
 
 ```bash
-claude mcp add schulcloud-stdio \
+claude mcp add --scope user schulcloud-stdio \
   -e DATABASE_URL=postgresql://schulcloud:schulcloud@127.0.0.1:55432/schulcloud \
   -- node --env-file=/absolute/path/to/.env /absolute/path/to/dist/bin/stdio.js
 ```
 
 `--env-file` keeps the JWT in `.env` rather than copying it into the MCP config.
-The HTTP form has no such option — its header holds the token — so leave it at
-the default `local` scope, which writes to `~/.claude.json` rather than a
-`.mcp.json` that would be committed.
+The HTTP form has no equivalent — its header holds the token — so that token
+lives in `~/.claude.json`.
 
 Check it:
 
@@ -66,6 +72,10 @@ claude mcp list       # schulcloud: http://127.0.0.1:8080/mcp (HTTP) - ✔ Conne
 claude mcp get schulcloud
 ```
 
+If `claude mcp list` does not show it, the usual cause is scope: run
+`claude mcp list` from the directory you added it in, and if it appears there,
+re-add it with `--scope user`.
+
 Then just ask: *"which courses am I in?"*, *"what's due this week?"*, *"find the
 material about Verschlüsselung"*. Inside a session, `/mcp` lists the servers and
 their tools.