MCP server listed as "Connected" in claude mcp list (CLI) but missing from session's connector list in Claude Desktop

## Description

A custom HTTP MCP server is shown as `Connected` by `claude mcp list`, but it never shows up inside an active Claude Desktop session — the session behaves as if the server doesn't exist at all.

## Steps to reproduce

1. Configure a custom HTTP MCP server (in this case named `eclipse`, pointing at `http://127.0.0.1:8642/mcp`).
2. Run `claude mcp list` — it shows:
```
eclipse: http://127.0.0.1:8642/mcp (HTTP) - ✔ Connected
```
3. Start/use a Claude Desktop session.
4. The `eclipse` server's tools are not available in the session — not in the loaded tools, not in the deferred-tools list surfaced to the model, and not listed by the session's own connector-status check (which does correctly list other configured servers, e.g. Playwright, Claude Docs, scheduled-tasks).

## Expected behavior

A server reported as `Connected` by `claude mcp list` should be usable (or at least visible/listed) within the Claude Desktop session.

## Actual behavior

The server is silently absent from the session entirely, with no error surfaced to the user.

## Environment

- Claude Code version: 2.1.235
- OS: Windows 11 Pro 10.0.26200
- Rebooted the machine before starting the session — issue persists (rules out a stale/zombie process on the local MCP server).

Reproduce the configuration with the `eclipse` HTTP MCP server at `http://127.0.0.1:8642/mcp`, then compare `claude mcp list` with the active Claude Desktop session's connector-status, loaded-tools, and deferred-tools output. Trace why a server reported as connected is omitted from the session, and consider the issue done when it appears in the session or reports a visible connection error.

Written by the indexing model from the issue text.