I have an Osprey Daylight pack. Enough room for 2 Seni Super Plus, a refillable wipe pack, small powder a new disposal bags. It can also fit a hydration bladder which is great.
I guess I elect to call mine a computer bag, no one needs or will know what is in there, it works well my sister-in-law got it for me. It can be packed with enough protection for a day outing
I use a messenger satchel atm. It was made by a company called dad gear and it was great for carrying my supplies. Unfortunately they went out of business . Been thinking about a tactical type cross body bag like the one you have pictured for my next bag as well.
I just keep a gym bag in the car, and extra supplies in my desk at work. I usually wear a sweatshirt or windbreaker and can hide a brief under my arm inside that while going from the car to a bathroom, if I have to change while out and about.