Arm designs cores that are being used in processors. Those are therefore Arm based processors. Apple is one of a very few companies that have a license to develop their own cores using the Arm instruction set, i.e., their cores are Apple cores and therefore their processors Apple processors.
Apple doesn't use ARM cores. They have a license of the instruction set that they use to develop their own cores, so technically they aren't ARM cores, but Apple cores that are using ARM instruction set.