Most employers aren't going to pay you more for opting out of insurance. Many employers would try to keep your hours low so they don't have to offer you insurance.

"When you say it like that, you are describing an employee not an independent contractor."

This doesn't make sense to me. She has more autonomy than an employee would. They give her that level of autonomy by not making her an employee. She does not get a W2.

"But your layer comment, about setting her own rates, makes me think that she is working as an actual freelancer, not as a gig employee."

There's a lot of overlap between those terms, except replace employee with worker. I would say she's a freelance gig worker. Freelance tends to mean that you are just independent. You can work on an individual project for a company and move on, but it's usually larger pieces of work. As a gig worker, I see it as being more of a repetitive task for various customers. You can be a freelance driver and work for both uber and Lyft, or even uber and a taxi company. The gig part is that they are always individual tasks you get paid for - each ride. This is very similar to what she is doing.

So I see gig workers as freelancers, but not all freelancers are gig workers.