Should an in-home caregiver be employed as an independent contractor or as an employee? My daughter works regular hours, with pay set by the caregiver.

This depends, an in-home caregiver can be classified as either depending on the situation and type of control the person she is performing the services for has. An employer controls what will be done and how it will be done.  If she is told, when, where and how to do her job, then she is likely an employee. If she is allowed to decide how to do her job, the lines start to blur. 

If she believes she is misclassified as an independent contractor when she should be an employee, you can file form SS-8 with the IRS.  This can take up to 6 months for them to process and give you a determination.  https://www.irs.gov/pub/irs-pdf/fss8.pdf