When caring for a newborn or toddler you do not need a diaper bag but you should have some sort of bag such as a backpack to keep spare diapers, clothing, wipes etc in so that you can have all the supplies handy when it comes time to change your kids diaper.
If you're gonna just be out for 1 or 2 hours then you may not even need to bring any extra diapers because diapers today are absorbent and can hold enough urine for a few hours without any issue.
My 2 year old boy went 4 hours without needing a diaper change in public when we went to a fair and I forgot to bring extra diapers.
He was okay for the 4 hours we were there and he was barely wet and didn't poop.
He didn't even feel wet or mind being in the wet diaper for the 4 hours.
He was in the same diapers for 12 hours while sleeping so the 4 hours is nothing with the new Pampers diapers he was wearing.
Pampers cruisers diapers can hold a lot of urine and not sag or feel wet to the kid wearing the diapers.
Any bag can be used as a diaper bag.
The diaper bag is just needed to carry diapers, wipes, bottles, toys etc or any other things needed to change your babies diaper.
I also recommend keeping a few outfits available to change the baby or toddler into should their diaper leak on the clothes.
I've always kept a few sets of baby clothes in a diaper bag which I used a backpack for which held more than a diaper bag could hold.
I kept a pack of diapers in the car as well so that I had enough diapers in case we got stranded somewhere and needed more diapers.
A good backpack is a better option for a diaper bag so you can put extra clothes in the diaper bag as well.