Most of my international travel has used one bag. In many cases I use hacks to avoid even paying the carry-on fee, and very rarely I'm caught and have to pay for the carry-on. But for my last long trip, I checked my bag, and carried a small bag onto the plane. I wanted to be able to transport stuff they won't allow as carry-on, and it was just less hassle to check it.
My favorite bag is a small soft roll-top backpack with a waist strap. Because it's a roll-top it can expand or be packed down tight, and it's narrow so it ends up fitting more places when it's packed tight. Outside straps and mesh water bottle compartments let me carry an extra pair of shoes (or water bottles). One exterior zip pocket, one interior zip pocket. I think one version of this bag has a padded laptop sleeve with zip, but I don't really need the feature. And I found one that has an internal frame which makes it more breathable and easier to carry when heavily packed.