Welcome to the forum!
Inkscape has an extension for creating jigsaw puzzle cutting files.
https://inkscape.org/en/~Neon22/%E2%98%85lasercut-jigsaw But as far as I know, it only makes rectangular puzzles. Here's one tutorial, which also uses GIMP in the process. But as far as I understand, it can be done without GIMP too. I think there is another tutorial somewhere, but I can't seem to find it.
When you wish for more templates, is the template the picture which is created when the puzzle is finished? Or is the template the pattern of pieces?
One thing which I find missing, both with Inkscape's extension, and with your example, is the lack of random variations, in both the shape of the pieces and the layout of the pieces. I've enjoyed putting together jigsaw puzzles when I was much younger, and I undestand that having identical pieces actually can make the puzzle harder to complete. So that some people might enjoy small puzzles, with nearly identical pieces, where the image is almost a solid color, or some repetitive pattern. Those can be very hard puzzles.
But I tend to enjoy puzzles with a very large number of pieces, which are pretty small, but where each piece is unique, and only fits on its other edges with certain pieces.
Anyway, I might play with the site you provided, and try to come up with some other templates. Especially if it allows for adding some randomization!
Thanks for sharing