general relativity - Is an atomic clock itself affected by gravity?

Sometimes I read that only time flows at different rates in different conditions when atomic clocks shows a different time compared to atomic clocks at altitude. But sometimes I read that an atomic clock ticks slower when gravity is higher.

Now the question is whether the number of cycles per second of a cesium atomic clock also slows down at higher gravity or that only the second is (also?) longer with higher gravity so that there is no difference?