The hour hand jumps, the GMT hand swings around on the Seiko. So def a caller and not a pilot. For me though it’s perfect because the GMT hand I keep constantly in one time zone and when I travel around I jump the hour hand for the time zone I am in. The rotating bezel I keep for home time when I am traveling.
What you described is the same as the Tudor, and same as the newer Rolex GMT's, which are called "Flyer". Meaning the hour hand is quick-set / jumping. It's called Flyer because like you said, it's convenient for traveling to be able to quickly set the local time by jumping the hour hand and not stopping the movement.
"Caller" is what most budget GMT's use, where usually the GMT hand is set by jumping it an hour at a time, after the local time and date have been set by stopping movement. It's called Caller because it's geared towards keeping the local time the same, and only adjusting the GMT hand or bezel to track someone else's timezone (you might want to call).
It's pretty sweet that the Seiko has the Flyer style, typically that's found on much more expensive watches often with in-house movements.
I use mine in the exact same way as you (jump local hour when traveling, leave the GMT hand set to actual GMT/UTC, and rotate the bezel if I want to track my home time or someone else's time. I find rotating the bezel so much faster than trying to adjust the GMT hand.