What exactly is a caregiver contract?

Elder law attorneys ARE a viable source for issues like this.

The nitty-gritty detail they forget to mention on web sites: Medicaid will not consider a lump sum payment for a family caregiver as a legitimate expense (rather than a gift), unless the lump sum is set aside in an annuity or trust account and controlled by a third party such as attorney, who doles out the $$ to the family caregiver a little bit at a time. Not for me, but maybe works for your situation.

Our social service office in our county was great. You can apply through a program called Public Partners & be paid through the govt. Let them handle the paperwork. You need a background check, that’s about it. You get paid taking care if your loved one in your own home!!

https://www.caregiver.org/personal-care-agreements
"A personal care agreement has three basic requirements for a person to pay a family member for care:
--The agreement must be in writing.
--The payment must be for care provided in the future (not for services already performed).
--Compensation for care must be reasonable. This means it should not be more than what would be paid to a third party for the same care in your state or geographic area. Tasks performed should match “reasonable” or “customary” fees typically charged for those services.
 
A properly drafted personal care agreement will contain:
--Date the care begins
--Detailed description of services to be provided, for example, transportation and errands: driving to medical, dental, adult day care, and other appointments, food preparation
--How often services will be provided (Allow for flexibility in care needs by using language such as, “no less than 20 hours a week” or “up to 80 hours a month.”)
--How much and when the caregiver will be compensated (weekly or biweekly)
--How long the agreement is to be in effect (The agreement should set time, such as a year or two years, or even over a personʼs lifetime.)
--A statement that the terms of the agreement can be modified only by mutual agreement of the parties in writing
--The location where services are to be provided (home of elder/adult with disabilities, caregiverʼs own home, other location. Allow for the location of the care to change in response to increasing care receiver needs.)
--Signatures by the parties, date of the agreement

 https://www.aarp.org/caregiving/financial-legal/info-2019/personal-care-agreement.html

https://www.agingcare.com/articles/personal-care-agreements-compensate-family-caregivers-181562.htm

Here are some websites about Family Caregiver or Personal Caregiver.  Copy and paste to your web bowser.  There are many, many more websites that have examples of caregiver contracts.  Each state has different regulations regarding Family Caregivers.  It is usually best to have an attorney write the contract to make sure that it is legal and that it meets the tax laws and laws governing the care of the elderly.

That's the thing... there is no exact definition of a caregiver contract because it is for you to define the job, pay, etc with your care recipient. But, if your loved one has the funds to pay you for care, it is a legal way to use their assets to do so. Here's an article with a sample.
https://www.agingcare.com/Articles/personal-care-agreements-compensate-family-caregivers-181562.htm