Hi Chouby (I've not yet figured out the F of your real name that I saw in the wp plugin list ;-) François?)
Adam has finally solved it and if you hear what was the problem you will probably think 'what a waste of my time'. Sorry for that, but shame aside I thought I report back it might help others in the future.
Pagelines Framework has a separate settings area in the edit page window to be able to do customizations also on a page-by-page basis. This area has a 'save settings' button apart from the Publish button in the WP page edit view.
I started a translated page with the Polylang widget in the edit sidebar and everything looked just fine. Settings were filled in and looked allright. The silly thing was that I had to re-save the Anything Boxes settings on the new page. This was unexpected.
After doing that Polylang can normally differentiate between Dutch and English posts.
So thank you for all your help and thinking with me. It brought it to a solution.
Jan