It looks like old patches managed to solve this issue for outdated versions → of this module. It would be great to have them ported to the current version (what unfortunately I don't have the knowledge to do).
I am not sure if this is related, but I tried to edit captions using Views with Editable Views module. When I update a caption of one image field, the caption from the other image field in the same node is deleted.
Thank you for your suggestions. In the meantime I've been taking a look at the Image Field Caption module, it seemed like a good and simple idea, I even installed it and quickly converted my text fields to caption fields. But I'm certainly going to investigate the possibilities of converting images into Media.
Thank you John_B for the reply. I randomically changed my image styles settings and fortunately the cron revived!