I think I have sorted out my problem, the following things are worth checking out for anyone with a similar problem. They may not solve your issue, but it’s worth a try.
1) I learned from another post here that Facebook caches images. So if you don’t get the right (or any) image the first time for a given post, no amount of reprocessing that post will change anything. You may make changes to WordPress that means the right image is now being submitted, but Facebook will ignore it, as it thinks it already has an image.
2) This can be overcome by using this debugging tool:
https://developers.facebook.com/tools/debug
Enter your wordpress post URL in it, and it will analyse it as if processing for a link (but without actually posting to Facebook). The great thing is that in doing so, it clears the cache, so you should run this after tweaking anything. Also it gives you some hints on what is going wrong.
3) Which lead me to the source of my problem. The above tool told me I had duplicate “og:url” headers in the post. As it uses the “og:image” header (which was also duplicated) to identify the image, this was the cause.
Why were they duplicated? Because I had two Facebook plugins: one to automatically forward my posts to my timeline as links; another to add “Link” and “Share” buttons for visitors. Seems they will not co-exist. Removing the “Link/Share” plugin solved the duplication.
4) And finally, my default image was a “normal” avatar/icon – 96 x 96. The debug tool told me that og:image sources need to be greater than 200 x 200 pixels (and experimentation showed me this does not mean 200 x 200 but greater than 200 x 200!). Otherwise, it ignores them.
So… I fixed my problem by only having one Facebook plugin, increasing the size of my default avatar/icon, and clearing the cache using the debug tool. I am now getting images in my link posts!
