The really weird part to get your head around here is that in case of relativistic differences, it's actually time that's changing. So cesium still has the same frequency, the same number of events per second, but their second is not the same as ours.

When people talk about clocks running faster or slower, it muddies the topic a little because it makes it sound like a fault of the clock. The clock is accurately measuring the time it experiences.

My favourite trivia on this topic is that this is not hypothetical. If the clocks on GPS satellites kept earth time (eg, without accounting for relativistic drift), the entire GPS system would accumulate 10km of error per day. This is a genuine thing with measurable, real-world effects.