Rental Agreement Backdating?

When I took my dad in, due to several health conditions that he was unable to care for alone and didn’t want to move to assisted living, I put together an estimate of costs. Things like cable (I stream), groceries, added expense of electric, water etc and then a modest amount of “rent” keeping it well below the cost of living in my area - a family deal so to speak. I’m now learning more and worry that I should have had a rental agreement in place for years prior otherwise, should the time for better care be a necessity, he could be penalized for “gifting” me rent. Can I backdate a rental agreement if he signs it? Do I need a notary or anything like that? I’m not trying to defraud the system just want to make sure him moving in with me and paying a very fair rent isn’t going to be perceived as a gift. To be fair I had to pay rent if I was going to stay at home as an adult and he would have never moved in without this. I should also mention he has a modest savings account that I have helped him invest and profit from but I don’t cover any of his bills as SS and retirement is enough that he has spending money and all needs taken care of. Should I still register as his caregiver?