If the arrows are supposed to show the direction of proper acceleration required for a body to keep the same spatial position on the plane, their directions are backwards. The arrow directions shown are the directions of geodesic deviation due to tidal gravity, i.e., the directions in which neighboring geodesics will move relative to each other. The direction of proper acceleration required to keep neighboring worldlines from deviating will be opposite to the direction of geodesic deviation.Figure 2 shows a sketch of the map seen "head on" that was sketched in perspective in figure 1.
I'm not sure this is a matter of convention, unless you are also treating the term "weight" as a matter of convention. In the usual usage, "weight" is a force and its direction should be a direct observable, which must be describable by an invariant. "The direction the object would move if it were force-free" seems more like geodesic deviation to me.The convention I used has the weights point in the direction the object would move if it were force-free.