> Isn’t the second defined as a specific number of cesium transitions?

Yes

> How can anything …

So your cesium counting device will fauthfully provide such a count and depending on their altitude it will be at different rates.

Both clocks are each experiencing time at the usual one second per second but gravity dilates spacetime.

Locally, a second is always a second, but from everywhere there is no such asbsolute, just as there is no universal "now".

I don't think this changes the way the second is defined. Rather, that definition describes some theoretical ideal where the cesium transitions are all perfectly equally spaced over the course of the second.

I think this new clock is simply able to generate more precisely spaced ticks than those of a traditional Cs clock. Less jitter and variation in the timing of those ticks. Similar to how a one-hour water clock or sand timer's runtime will vary between "transitions", but a one-hour quartz stopwatch timer is much more regular. I could keep going, but I'm already out on a limb so I'll stop before my own uncertainty rises too much.

(Edit: I read the article. I don't think my words above are correct.)

The idea here is to have a clock that actually ticks faster and to redefine the second in terms of that new, faster ticking speed.