In my years of renting, upon signing the lease, the deposit is due (usually matches monthly rent payment), if you move in halfway through the month, then prorated rent is due on your actual move in date. You shouldn’t be paying next months rent up front before you pay the deposit and the prorated rent. I’ve never heard of that.

But your original lease will say right on it what the refundable deposit was and what the monthly rent rate will be, and what your prorated rent amount is(if applicable). And in this situation it would apply.

You shouldn’t be renting or paying anything if you never signed a lease. ALWAYS keep the lease.