The effect you are describing was discovered theoretically by considering a though experiment (at least that is how many texts present it). Ampere's law relates the line integral of the B field around a loop to the number of currents that pass through any surface bound by the loop. The mathematical theorem is Stokes' theorem.
When there is current in a wire there is a B field circulating around it. The integral of this around a circle enclosing the wire is proportional to "I" poking through the surface bound by the curve. The thought experiment goes something like this. Consider a charging or discharging capacitor through a wire loop with some resistance. For simplicity consider discharging. The circuit seems to be a closed loop and while current is flowing there would be a B field around the circuit. Now, calculate the circulation of B around a circular loop that surrounds a segment of the wire and consider the flat plane surface bound by it. Clearly the current in the wire pokes through this and Ampere's law holds. But mathematically speaking to should hold for ALL surfaces bound by the curve. Now consider a surface that is bound by the curve but morphed to pass through the space between the capacitor plates. Clearly there is NO current poking through this surface but there is a circulation of B along the curve! This lead to the realization that the changing E field acts as current and creates a B field. Notice the choice of words "acts like". The changing E field makes up for the discontinuity in I and the two work together to generate B.
This was not an easy thing to arrive at. I agree with the comment that it is perhaps a poor choice of name but it has stuck over time.
To answer your question directly, yes it is possible that with air or some other medium in the space a small current is set up but this is really more of a polarization effect that changes the capacitance. It does not account for the phenomenon. Also, in principle the effect has to be there in a vacuum and your proposed explanation would break down in that case.
(sorry for not posting a figure. The description of Stokes' theorem and Ampere's law require 3-dim, at least).