I think I may have figured this out - perhaps this will help others too.
I had the following setting ticked:
Mark media I tweet as containing material that may be sensitive
But I didn’t have the following setting ticked:
Do not inform me before showing media that may be sensitive
This seems to mean that my twitter card images will not show when the links are tweeted by my account unless the viewer has chosen to view sensitive media previously (they do not get the option to show the media, as they do with a normal tweet). As I had not chosen to view sensitive media, the images on the cards were not shown to me, even though I had tweeted them!
The cards that were working for me (even though my settings required me to be informed before seeing sensitive media) were tweeted by other users who (presumably) had not marked their own tweets as sensitive!
If I tick the Do not inform me before showing media that may be sensitive setting, my card images now show up for me.
It may be worth adding something to the docs to say that another possible reason for a twitter card image not to show might be the Mark media I tweet as containing material that may be sensitive setting in combination with the Do not inform me before showing media that may be sensitive setting…
Thanks for getting back to me, though, I’ve been pulling my hair out on this one!