The deposit itself is not a liability. The deposit account is. There are many ways to get money into a DDA. The bank will never give you your original paper currency back.
I agree with everything you say, but at a high level they "owe" you the money that you deposited. Which is a liability. They (generally) are not allowed to just go spend your money on anything they like (in the same way you could if you got a business loan, which is an asset and a liability), but it is prescribed how they can create assets with the deposits.
The deposit itself is not a liability. The deposit account is. There are many ways to get money into a DDA. The bank will never give you your original paper currency back.