What form do I give a caregiver that we directly pay?

If the caregiver works in your home, she is a household employee. If you pay her more than $2,600 for the year (for 2023) you have to give her a W-2 at the end of the year and you have to file the W-2 with the Social Security Administration. See IRS Publication 926, Household Employer's Tax Guide, for all the details of what you have to do.

If the caregiver works in her own home or other location that she provides, she is in business for herself. You are a customer of her business. You do not give her any tax form.