When I try to share links from this URL on Facebook, I can only choose one image as the preview:
But when I try to share links from this URL, I can choose between five thumbnail previews:
Any ideas what the difference is?
Facebook doesn't seem to provide details on how it chooses those thumbnails. One difference between those two sites is that the one where you only get one image has a sidebar on the left, which means that the posts and the pictures are further down the page. It may be that the script that looks at the page doesn't find the picture near the top of the .html file and so gives up.
You could try a different Wordpress template without the sidebar on the left. Alternatively it is possible to have the sidebar further down the file than the posts, but still display in the same position, but that would mean editing the theme by hand.