I don't think so. If it would be as you said the Bf 109E and other kites had the spinner painted in the way as well. Also what about these spinners painted without the white marking? IMHO, the white "quarter" was for warning actually. It was not only for the maintenece crews but also for birds. In other words when the prpeller was rotating the white marking ( no matter if it was the white "quarter" or the spiral ) caused its moving better noticed, especially that Germans didn't paint the prop blade tips painted yellow or white.
And yes the pic you attached below shows the training variant of the Bf 109. But it doesn't matter , the rule was for all late Bf 109s.