An air break is an opening in a piping system. It is arranged so that with normal water flow there won't be any water leaking out, but if/when water is drawn backward through the source pipe it draws in air at the opening (the "break") rather than pulling water up from the drain pipe. And you're right: an air break can be a concept (the simple arrangement of the drain tube being held in a larger pipe), or it can be an actual physical thing (like the gray plastic device in your picture).
Whatever way the air break is realized, it doesn't need to rise higher than the top of the washer cabinet; it only has to rise a little higher than the water fill level in the wash tub. If the whole works is much higher than necessary simply because the standpipe is so tall, then just cut it down lower.
The force of water pushing through the drain tube can case the tube to lift, so it's important to secure the tube well just as you said.