In general, 'conductance' (or 'resistance') per cross section (or width for 2D systems) of the conductor is a well-defined quantity and has a physical meaning for any type of materials, material structure (heterostructures, interfaces, contacts) or device. The 'conductance' concept depends on neither the dimensionality of the system nor the type of the electron transport (ballistic, diffusive, coherent or incoherent).
'Conductivity' is a well-defined, physically-meaningful quantity only if the conductance obeys Ohm's law, i.e., conductivity = conductance Length / Area = constant, i.e., that the conductance scales inversely with the system length, also meaning that the electron transport is no longer ballistic.
For 2D systems, the definition of conductivity you mentioned in your original post is totally fine, provided that the conductance scales inversely with the system length.
