if you only need light protection, here's my advice:
go to the box store and get a backpack with two main compatments. (in addition to the small front compartment) They do make them, I own one. If you're going to be out traveling, it's not too unusual to wear a backpack. Put a light windbreaker jacket in the main compartment maybe along with other things like an umbrella, sunscreen, map, cell phone power bank, bugspray, etc to make it perfectly normal/legitimate to have the backpack with you. No one will question it.
Obviously the other compartment is to carry a change or two of diapers and any small supplies you need. Northshore has really handy small pouches of wipes that are ideal for a light diaper bag. I would personally recommend removing the pull on the zipper to your diaper bag area of the bag, to make it impossible for someone to open by accident if they want to borrow something out of your bag or put something in it if you leave it unattended. I do this with my big work lunchbox's top pouch where I carry a single change, after it got opened by accident by someone looking for a tool they knew I was bringing to work. You should still have little trouble operating the zipper without the pull. Park the "public" compartment's pulls in the middle at the top where they're easily seen, and park the private zippers all the way at the base where they're hard to spot.
My current best recommendation for light-diaper-on-the-go is Tykables Little Builders. They're quiet, thin, soft plastic shell, good velcro tapes, very comfortable (rear AND front elastic), durable enough to wear all day without splitting (though you may shred the padding inside a bit, to be expected), and very easy to put on in public due to the easy-adjust velcro. (they're a wonderful daytime diaper!) Betterdry are nice but the tapes would worry me a little (they slip after a few hours on the move) and they're a bit thicker than I'd like for a ot of walking. Definitely go with them instead though if you need higher capacity. Betterdry also has incredible leak guards so they are a good choice for overnight.