You can control what is being displayed on transaction listings (Account register, Budget register, reports). This is controlled by whichever field in Transaction layout settings is on top. If you have Payee field placed above all other text-based fields, if MoneyWiz can only show you one piece of information, it'll be Payee. If you place Category above all other text-based fields, it'll show you category.
Please note that a field does not have to be on the very top of the transaction layout settings - just above other similar fields.
This image should explain it better:
Notice how Category being on top of other text-based fields (but not necessarily on top of everything) caused it to be the first thing to appear on transaction list. If we dragged Payee to be above Category, those would change places on the transaction lists too.