In the United States, we call your first item a "shopping cart." The second is "a shopping basket" or "a hand basket."
We do not use the word "trolley" the way the British do.
In a restaurant, a wheeled device for showing diners a selection of desserts is a "dessert cart." In an office, a wheeled device to allow a computer to be moved easily from one place to another is a "computer cart."
In the United States it is used mostly to describe electrically-powered streetcars that get their power from an overhead wire--and there are very few such systems left, so we hardly use it at all.