Don't be confused - 60 days notice is always required under your lease if that's what it says. You only gave 30 days notice, so you're in breach of the contract and they can charge you the additional month of time even though you moved out.
The fact that you paid for 12 months plus 1 means nothing really - you lived there that long. You didn't give 60 days notice as required, thus you're being charged for the other 30 days you were required to give them notice prior to moving out.
The first term of the contract was 1 year, then it extends one month more at a time. Either of you (tenant or landlord) could give proper 60 day notice to the other that you were going to terminate the lease in 60 days, and then at that time it would be over. The month to month is the extension after the first term - sometimes leases are written to be for a certain period, and then renewed automatically for another same length of time. Residential leases (unless otherwise specified) got to month to month. But the length of the lease has nothing to do with the notice period.
Bottom line - you owe what the landlord says you do.