"Crossover" was essentially "minivan without sliding doors."

4x4 generally means "you have a second gear selector to choose 2wd or 4wd Hi or 4wd Lo," whether that selector is a mechanical lever or a switch.

"Full time 4wd" and AWD mean that the car is always driving all four wheels, there is no selector to switch to 2wd, and usually with the primary being front wheel drive, with a teeny driveshaft sending a small percentage of the power (like 20% or 30%) to the rear wheels (though that may change based on traction control sensor inputs).

I apologize for the length of the previous sentence.