I wonder if cabinet is American English for cupboard in UK English

I would call the following cabinets and the counter in the middle an island.

What would you call the following, Shelf? The owner calls it cupboard. Do Americans call it cabinet/closet? To me, a closet is usually hidden in a small room (walk-in closet) or recessed into a wall (built in)

I find it hard to differentiate between cupboard, cabinet, closet and shelf.

Thanks in advance!

In the first image, these are kitchen cabinets containing cupboards, with a peninsular work-surface that has cupboards below. An island work surface is completely detached and standing on its own.
The second shows a cupboard.
A closet is generally a small enclosed room.
A shelf is a board on which items are stacked. Books are housed on bookshelves.

A cabinet is technically 1 a cupboard with drawers or shelves for storing or displaying articles. 2 a wooden box or piece of furniture housing a radio, television, or speaker. A person who makes furniture is a cabinet-maker.
The definition of cupboard was originally a place where drinking vessels were kept, and has expanded to cover a piece of furniture that is used to store anything that is kept behind a door and on shelves.
The American answer is that those are cabinets, and the island has cabinets under it.

As for the second piece of furniture, it's not a common one here, so what people will call it will vary. An enclosed bookshelf is one possibility, or possibly a cubboard.

I believe Nona said recently that in the UK, a cabinet is in the bathroom, and is what we call a medicine cabinet.
Students: Are you brave enough to let our tutors analyse your pronunciation?
I'm sorry but I still don't see concrete discriminant features that set cupboards apart from cabinets, functionally or structurally. I should have made my questions clearer. Both pieces of furniture have shelves and drawers. One thing I've observed is that cabinets are built in and the cupboard in the second image is not.Is that the only discriminant feature?

Thanks in advance.
A cupboard is one form of cabinet, a piece of furniture with shelves, cupboards, or drawers, which is used for storing or showing things. A display cabinet will have glass doors and sometimes glass walls and shelves as well; a medicine cabinet is used in a bathroom and often has a mirror front; kitchen cabinets can contain shelves, cupboards or drawers.
Cabinets do not need to be built in, and generally are not.
A cupboard is one form of cabinet, a piece of furniture with shelves, cupboards, or drawers, which is used for storing or showing things

Feebs, what's the italicized phrase describing, a cabinet or a cupboard?
Students: We have free audio pronunciation exercises.
A cabinet.
Personally I think you can become too tangled up in trying to find a precise meaning in this situation. There are so many ways in which items of furniture are identified.
However, basically a cupboard is a piece of furniture with a door or doors, sometimes with shelves and sometimes without, sometimes small and sometimes tall, in which are kept small domestic items [china, glass, kitchen utensils, pots and pans, and so on]. Cupboards are sub-categories of cabinets.
You're right. I'm really confused. I guess this is one of those
things you have to learn from experience.
Thanks for your time.
In the US, attached kitchen storage with doors as shown in the slcc.edu photo are usually referred to as cabinets. You are correct in calling the counter in the middle an island. The kandaworkshop.com piece, because it is not attached to the walls, is considered by most to be a cupboard, an older term dating back to times when kitchen furnishings were minimal and not built-in. Open storage (no doors) are typically referred to as shelves, and closets in kitchens are usually considered pantries.

One caveat: some people do call kitchen cabinets "cupboards." This is probably a regional phenomenon.
Teachers: We supply a list of EFL job vacancies