The reason bettas don't need as much space is because they don't need to move around to breath. They breath from the surface. having too much space can actually tire them out as they will have to stay near the surface more often than usual and don't get the "rest" periods as often.

Yeah I can't remember where I learned the filter thing.... but if you ever watch a betta fish you will notice that they often will rest on a plant or other things and be perfectly still. no swimming or anything... they need that rest and sometimes the movement from a filter can move the water so much that they can't rest that way.