British biscuits are closer to the original biscuits. The word biscuit means "twice cooked" (cuit is still French for cooked). This refers to the kind of biscuits that were taken on ships. They are made by first making a big cake, then chopping it up and cooking the pieces a second time to dry them out.

Such biscuits are still popular with Afrikaaners in South Africa.

Cook and cake have the same origin. Cookie just means "little cake" and therefore is completely correct in the way Americans use it. It just has a different origin to biscuit.