Yes, in most everyday situations, urban trash cans with dome tops really do keep rain out better than open-top ones. The dome acts like a small roof: it sheds water away from the opening and reduces the amount of rain that falls straight into the bin. Open-top cans are basically buckets for rainwater, so after a storm they can fill up with soggy waste, get heavier for collection crews, smell worse, and overflow faster.
That said, dome tops are not magic. A lot depends on the design. If the side opening is too large, if the hood is too shallow, or if rain is driven sideways by wind, water can still get in. Some dome bins also collect moisture through condensation, especially in humid weather or when bags are not fitted properly. If the bin has no drainage holes or the liner traps water at the bottom, a dome top can still leave you with a wet, smelly mess.
In real urban use, the best performers tend to have an overlapping hood, a relatively small side opening, a sloped top, good ventilation, and drainage that lets any trapped water escape. Frequent emptying and well-fitting liners matter too. So the short answer is: yes, dome-top trash cans generally keep rain out better than open-top ones, but only if the details are designed well. A bad dome-top bin can still let water in; a good one clearly has the advantage.