Electrons flow clockwise through the circuit drawn. You could say that electrons accumulated on the anode are lost to the wire on the right of the anode, and that electrons flow onto the left of the cathode so that the cathode gains electrons. That is the more natural interpretation if you think of the discharge tube as a component in a larger circuit.
But I agree, it is a little ambiguous. Which is why a better terminology would be to just refer to the "positive" and "negative" plates, and reserve the terms "anode" and "cathode" for instances where we actually have oxidation and reduction reactions occurring at the electrodes, in which case the electrochemistry is completely unambiguous.
Unfortunately, the ambiguous terminology is still quite common!