I think of the grain in a tree in relation to quarter, rift or plain sawn. When it comes to cutting/serving boards (may not be correct but how it was taught to me) I am just using the term to describe how the cutting board is being constructed in relation to the wood orientation and the grain orientation really is more tied back to how that board was cut. I find though that flat sawn and face grain produces a really nice serving board, edge and end are of course better for cutting on but for the beauty of the board that's the combo I like.

If the board is laying flat face up/down that is face grain. Cutting several flat boards (usually different color hard woods) into several varying widths and gluing up will produce what I consider a face grain board. might have a couple of 'edge' flipped strips for accent. If I cut a flat board into say 1 1/2" or 2" inch strips and then flip them on their sides and then glue those up and shape into a board it is 'edge' grain. Slice that horizontally and flip on end then its 'end' cut as the end result is really cubes of the original board on end. Based on your question and considering its coming from an old pro of 50 years I assume this doesn't ring true but right or wrong it always matches up with what's out there for sale.