Since the lamps are in series, the electric current through each is identical; all of the current out of one lamp is in to the other lamp; if there is a flow through one, there is a flow through the other. It cannot be that there is a flow through one and not the other (in the context of this simple model).
Thus, if the lamps are identical, their brightness is identical. This is all there is to it.
Regarding the distinction between electric current (flow of electric charge) and electron current (flow of electrons), the flow of electrons counter-clockwise is an electric current clockwise since electrons carry negative electric charge.
In an electrolyte, for example, the flow of both positive and negative charge carriers contribute to the overall electric current. The direction of the electric current is the direction of the flow of positive charge. Thus, a flow to the right of positively charged ions and a flow to the left of negatively charge ions contribute to the electric current to the right.