You won't see the exemption" until the second full tax year you are there. The amount they use is the purchase price. Google is your friend:
"Save Our Homes" Amendment and Its Effects - Rick Singh, CFA - Orange County Property Appraiser
Save Our Homes Amendment | Property Tax in Florida
This is better explanation:
"Save Our Homes" Amendment and Its Effects - Rick Singh, CFA - Orange County Property Appraiser
Example: On our TRIM notice the appraised value of our home went up $28,000 but the increased taxes we pay are only based on something like 3% or so of that difference. Took effect in year 2 with year 1 being the base year.