Part of this is a nomenclature problem: some call it against the grain and others callit it across the grain. Across the grain is a little better terminology. This also implies that the cut is mostly straight up and down, with respect to the cutting board.
You can also use a "bias" cut, which is still an across-grain cut, but adds interest to the eye appeal of a serving. Such cuts are made across the grain but at an angle to the board (not vertical). It has most of the benefit of short-cut grain plus the eye appeal.
Well said.