I've never worked out what the difference is when going forward, but the synchro coach insists that slidy back crossovers are called crosscuts and the lift the foot up variety are crossovers. I would say they were both crossovers and I just need to be told whether I'm lifting my foot or sliding (sliding gets power but he thinks lifty looks great if the whole team get a change in height at the same time).
But he's the only person I've ever heard define them, and I never normally hear anyone talking about crosscuts in the UK (and I'm sure it says crossovers in the moves test manual).