Nice video walkthrough
You do the "Race jibe" with front rear foot shifted forward on the way in through the jibe.
Basically there are two ways you jibe on a foil, the race jibe and the surf jibe.
Surf jibe is where you go through the turn exactly like you describe but dont switch your feet till on the way out almost on the new coarse.
Then you do the same steps, rear foot forward and forward foot rear - the faster the better, and eventually you can make it in a fast little jump.
Both jibes can also be performed in a little jump with the feet.
But usually a really fast dub-dub two step really fast foot shuffle is "safer" as you wont get huge crashes if you miss
And you can practice this when board on the water, learn to dub-dub faster and faster or even a small jump.
Can be practiced both on your foil or on a waveboard.
It is the same techniques whether you are foiling or on the water - except that when foiling you have almost no room for errors and it is super sensitive of course, so here a little or a lot of kitelift helps you get weightless for an easier foot change, and the kite timing is essential, a lot more than when on the water.
Those coming from surfboards having learned the typical surf jibe (feet switched on the way out) often start learning this on their foil too, but as so many do the race jibe some learn this instead.
I would say the race jibe is a bit easier for most
I can not do it myself, never trying though as not for me, but just personal what one likes and can.