Let's look at it this way

Facts:
1. The seller paid tax for the period of Jan 1 to Mar 31 to the county prior to the close of escrow.

2. The escrow closed on Jan 7 and you own the house since then.

Conclusion:
You are supposed to pay the tax for Jan 7 to Mar 31 because you are the owner, but the seller had already made payment for you. You should reimburse the seller - that what the $512.12 was for. You were NOT cheated. The 4th installment payment is irrelavent to above fact.