Take the simple horse saddle (z=x^2-y^2). At (0, 0, 0), there is no slope. However, it is not a minimum or a maximum point, as I can slope up (to, say (1, 0, 1)), and down (to (0, -1, -1)).
Take the term saddle literally. In order to saddle something, you need the higher points on some (opposite) sides to stabilize your body (as given by (1, 0, 1) and (-1, 0, 1) in the simple horse saddle), and some lower points (also on opposite sides) for your legs to go (as given by (0, -1, -1) and (0, 1, -1)). The "saddle point" would then be the point where (the center of) your body goes (i.e. (0, 0, 0)).
2020-07-12 16:30