Yes, any Cursor sub includes a separate tier of usage for Cursor/SpaceX first party models separate from usage of third party models.
I have a $20 Cursor sub still personally and would say that the usage for Grok models is much higher than what $20 gets you with Anthropic. I can't speak for the $100 plan but I would assume it scales similarly.
I rarely run out of my usage monthly even with the $20 plan.