I think "agent mode" is the default mode in Cursor. And I think that means that when you ask it to do something, it'll go make code changes in multiple files, create new files, run commands, etc, all in one process (but you watch it do its' thing and can approve/reject each change).
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But "it does the work directly" seems right.
in Cursor it's a specific mode.
Elsewhere, it's more like delegating passive tasks to AI. Lotta hype here around MCP servers, n8n, etc.
Regarding Codex, I would assume most people would use it alongside/in their editor to review and confirm the files it's creating/updating.