A practical approach is to use a normal backpack or shoulder bag and add a padded camera insert. That gives you the most discreet setup because the bag won’t advertise camera gear, and inserts are inexpensive and easy to move between bags. People often repurpose army surplus bags, everyday backpacks, cooler bags, or diaper bags for this.

For rain protection, many non-camera bags aren’t fully waterproof, so a simple backup is to carry a plastic liner or garbage bag inside and use it as a rain cover when needed.

If you want a ready-made camera bag that’s more understated than most, the Think Tank Retrospective line is one option mentioned by users, though a converted everyday bag is still likely to be the least conspicuous.

Given your priorities, a plain backpack plus padded insert is probably the best fit: discreet, lightweight, flexible, and easy to use as carry-on.