@David Jesse You are right. You have to have possession of the property on or prior to Jan 1 to homestead for 2020. You will be responsible for taxes (non-homesteaded) for 2020. These will be based on your purchase price. Remember, property taxes are paid in arrears. That means 2020 taxes will be due Jan 1, 2021. (You actually have until March of 2021 to pay, but many people actually pay early for a discount, November 2020 for example).
So in your case, between Jan 1, 2021 and March 1, 2021 you will have the opportunity to homestead for 2021. Your 2021 taxes, due Jan 1 2022, would be discounted with your homestead qualifications. If you miss the March 1 deadline, you'll have to wait till the following year. Depending on the previous homeowner's homestead history (how long they have lived in FL, widow, etc), your homestead discount could be completely different. Never look at past tax records to determine property taxes in FL. I recommend you call or visit the property appraisers office in your town once you've moved and get all the details from them. There are some things you can do to start the process early, but I don't think it changes how much you will owe.