Duel between lamps

Unfortunately, your assumption is not correct.

There is no change in the current flow. Due to the fact that the polarity of the voltage source has not been reversed, the electrons still transit the right light bulb before the left one.

Gegenüberstellung zweier Fotos mit vertauschter Reihenfolge der Glühlampen. Sie leuchten beide jeweils unabhängig von ihrer Reihenfolge gleich hell.

Depending on the construction type of the light bulb, it shines with a certain brightness at different current strengths. In this case, the oblong light bulb has got a spiral wound filament (the wire inside of the light bulb which glows while current is flowing) with a smaller diameter and thus is shining brighter.

Think about what this means in case of the current strength. In what way is it influenced while transiting the lamps?

a) The current strength does not change while transiting the light bulbs, therefore the certain brightness of the light bulbs does not depend on their position.

b) In the darker, round light bulb electric current is hardly used up. So the other light bulb shines as bright as before when it was the first to be traversed.