From https://code.claude.com/docs/en/remote-control :

    Feature-flag evaluation: DISABLE_TELEMETRY, DO_NOT_TRACK, CLAUDE_CODE_DISABLE_NONESSENTIAL_TRAFFIC, and DISABLE_GROWTHBOOK each
    disable the feature-flag evaluation that Remote Control availability depends on. Unset the variable
    wherever it’s set, in your shell environment or in the env block of a settings.json file, to use Remote Control.

I just discovered the same thing, that my local sessions were sent to claude.ai/code and stored there. I added:

    "env": {
        "CLAUDE_CODE_DISABLE_NONESSENTIAL_TRAFFIC": "1"
    },

to my .claude/settings.json file, and the /rc in the corner went away.