yes it is correct that it is "Udbetaling Danmark" that will be paying maternity leave benefits.
However since you are member of an A-kasse, and are entitled to unemployment benefits from the A-kasse, you should contact your A-kasse and tell that you are going to maternity leave.
The A-kasse notifies "Udbetaling Danmark", which sends you a letter to you with Digital Post (e-boks.dk). You digitally confirm that the information from your a-kasse is correct.
Normally you will get the same amount in maternity leave benefits (as you got in Unemployment benefits), however when you give birth (goes on maternity leave) your benefit will be higher, because you will have dependant child. As you mention it will be DKK 15.470 per month.
Anders Weber, www.a-kasser.dk