Frequency-dependent brightness fluctuations of radio sources, the so-called extreme scattering events (ESEs), have been observed over the last three decades. They are caused by Galactic plasma structures whose geometry and origin are still poorly understood. In this paper, we construct axisymmentric two-dimensional (2D) column density profiles for the plasma lens and explore the resulting ESEs for both point-like and extended sources. A quantity that becomes relevant is the impact parameter b, namely the distance of the observer's path from the lens' symmetry axis. We demonstrate its effects on the shape of ESE light curves and use it for a phenomenological classification of ESEs into four main types. Three of them are unique outcomes of the 2D model and do not show a characteristic U-shaped dip in the light curve, which has been traditionally used as an identification means of ESEs. We apply our model to five well-studied ESEs and show that elongated plasma tubes or quasi-spherical clouds are favoured over plasma sheets for four of them, while the remaining one is compatible with both lens geometries.