I believe the idea is that a single bright star in the frame (the guide star) is used for selecting the frames. The point spread function (PSF) is just going to be some function that describes the blurred shape you would observe with the detector for an input point source. You then select frames in which the guide star is well centered, compared to its overall distribution.
I think your guess on “sync-resampled” is correct. They increased the “resolution” by a factor of 4, so that when they realign the chosen frames to center the guide star, they can do so at a sub-pixel precision.
You may want to check out chapter 3 in the thesis, particularly section 3.5.3. The give a lot more detail on the process than you’ll be able to find in the paper. A well-written PhD thesis can be 1000x more valuable than the journal article it ultimately produces, because it contains all the specific details that can be glossed over in the final paper.