Do you know the difference between a cross over and 4x4 SUV?
A 4x4 SUV is basically a truck chasis/motor with a passenger vehicle on top.
It's powerful. It can tow. It can go off road.
A crossover (erroneously called an SUV in Australia) is a regular car chasis/engine, made to look like a mini 4x4.
Some have all wheel drive, but this should be considered as an extra feature, not a category changer.