I always thought 4th axis meant about a rotational axis over & above the 3 normal linear axis. But I think there is a more robust definition like below. I found this definition on the internet, so it must be true LOL. I guess you could have a machine with no rotary component but some other 'direction the cutting tool could move' and that might also be a 4th axis? (Although I cant think of one off hand).
The term “5-axis” refers to the number of directions in which the cutting tool can move. On a 5-axis machining center, the cutting tool moves across the X, Y and Z linear axes as well as rotates on the A and B axes to approach the workpiece from any direction.