From what I’ve read most French people call it pain au chocolat. Apparently some politicians tried to get protected status for the name chocolatine and failed.
It looks more or less identical to what my local bakery in France used to sell as 'mille feuilles'... I think it's a fairly broad term, but I don't think I've ever seen it with more than three main layers of pastry. Certainly no-one in France uses the term 'vanilla slice' to describe anything
And for pain au chocolat, don't get influenced by these inferior french speakers that called a chocolatine. You're absolutely right and there's absolutely required to stick with this ortograph, or you'll have a vast majority of French booing you
Singular form exist and is used more often.