"Shopping basket" or "shopping cart"

As to whether it should be "Basket" or "Cart", I would have said this was a localisation issue. The US seems to favour "Cart", but here in the UK "Basket" is more correct. eg. amazon.co.uk uses "Basket", amazon.com uses "Cart". In fact, I'd go so far as to say "Cart" is incorrect for a UK based website.

I think "Cart" or "Basket" is simply the shortened form of "Shopping Cart" and "Shopping Basket". You don't want to over use the word "Shopping" if its context is obvious IMO. "Shopping Basket" would be the title and/or perhaps used the first time it is mentioned in copy and "Basket" thereafter. eg. "Your Basket is empty", "You have 2 items in your Basket".

Shopping baskets and carts are physically very different.

Shopping Basket

A metal or plastic container with one or two handles, used to carry shopping in a shop.

Shopping Cart

A four-wheeled cart provided by a supermarket or other retail store for a customer's use in collecting purchases.

Graph created in Google Ngram Viewer.

It's interesting to note that Shopping Cart is more popularly used both in American and British English. Shopping cart is today's most popular choice. Probably due to the fact a cart can hold more items then a basket and most people show at supermarkets with trolley's. Baskets are for smaller purchases.

Usually, it's just "Cart", as in "You have 2 items in your cart"

This can be seen here, at the top right hand corner.