> Conversely, in the gentle field of Earth's gravitational field, if you hold an atomic clock in your hand a meter from you head, and an atomic clock in line with your head, then yes, you can see a tiny difference in the tick rates, on the order of nanoseconds per century or somesuch (I'm too lazy to work it out).

This is why GPS satellites need time correction. GPS satellites contain atomic clocks, and they are further from the center of the Earth than you are. Therefore they experience slightly less gravity than you do. Therefore their clocks run slightly faster than atomic clocks on Earth's surface. That's the General Relativity correction.

But GPS satellites also move a lot faster than you do, because they are in orbit. That makes their clocks run slower than clocks on the ground. That's the Special Relativity correction.

Both corrections have different signs but they do not cancel each other out.