Cross-through and push-pull are very similar. Basically the same, but cross-through does not include the rotation technique of the EC style.
On ice, yes, you need to stay very angulated and really finish your turns. Carving round turns controls speed, gliding between carves allows speed to build, which is why you want to do a quick cross-through to minimize time between carves. You're right that when you're very low you don't have room to cross-under, hence the need to rise up a little as if you're starting a cross-over, and then finish the transition with a cross-under. This is cross-through.
The point is to move as little mass as possible when changing edges. Move the light parts (board, boots, lower legs) under the heavy parts (the rest of you) instead of moving the heavy over the light. Of course, if conditions are good enough you don't have to do this, and then a big cross-over can be really fun - that's when you get that weightless feeling, or even catch air.