What are "fondi di carciofo" called in English?
(American English speaker here) To me, the picture of "fondo di carciofo" is a picture of artichoke hearts.

What you seem to call a heart, I call a baby with the outer leaves removed. Like this:



I don't think of that as a heart, I've always considered that a kind if a cheater thing since that's what you can buy frozen or marinated as "hearts". There is no, or very little, inedible choke: 



That's because the choke develops as the artichoke matures.

So if I asked for "baby artichokes", I would expect to get what is pictured above.

If I want what I think of as "heart", I would eschew packages labeled "artichoke heart", because I know that what they would actually contain (in the US) is trimmed baby artichokes.

I LOVE artichoke hearts, I try to get plenty of stem too. I'm talking about the bottom portion of the mature artichoke, after you pick off the leaves and scrape off the choke.

This:



Which just has a small part of the choke removed.

On Food and Cooking supports this usage of "heart": 



So, I guess "the bottom part of a mature globe artichoke, the base, what some people consider the heart and others call the bottom or even the crown, where the stem attaches" would be a way to say it in English and not be misunderstood. If, in fact, I am understanding the question.