After some more reading, I think I better understand the issue. I summarize my findings here, please correct me if something is wrong.
In the subcritical zone, there is a phase transition between liquid and vapor when the temperature and pressure lie on the phase boundary. The phase transition is characterized by the fact that liquid and vapor exist simultaneously in equilibrium.
The key experiment to show the critical point is to keep the liquid/vapor in equilibrium, while slowly moving to the right along the phase boundary (simultaneously increasing the temperature and pressure). At some point (the critical point), the two phases become identical and the resulting fluid is called supercritical.
Since we were moving along the phase boundary, the supercritical fluid appears when both the temperature and pressure become supercritical, which justify the standard definition of a supercritical fluid. The following 3D diagram from http://en.wikipedia.org/wiki/Phase_diagram is quite useful for visualizing the critical point.
However, the transition from liquid/vapor to supercritical is only clear when following the phase boundary. If one goes around the supercritical point, nothing visible appears and one goes smoothly from liquid to vapor. As a consequence, there is no clear physical delimitation of the supercritical zone, the definition is just conventional.
Moreover, if one were to seek a delimitation somewhere, the Windon line between liquid-like and gaz-like behavior (see http://www.nature.com/nphys/journal/v6/n7/full/nphys1683.html) would be a better candidate.