Hi. I'm native. 장바구니 is called as same as 바구니, 쇼핑백, and others. So 'Shopping basket' means 'a bag used when someone is shopping', but we don't always use '장바구니' when we are shopping. 바구니, 가방, 백, 쇼핑백, 비닐봉투, 비닐봉지 등등 다 쓰인다고 빡대가리 여러분아!!!!
These are the handheld ones, not the carts. 바구니 by itself is basket. 장 I imagine comes from 장을 보다 (to go grocery shopping).