The screenshot with the dragon flying above tells me no. It's what ever weather it was when you found the dragon.
I agree, if the weather in that screenshot is supposed to be special dragon weather, then it's pretty underwhelming for that, because it looks like normal weather to me.
In any case, why would dragons cause a change in weather? Do dragons just have the natural ability to control weather? Or does Alduin just change the weather whenever a dragon appears because he wants to lower the morale of his enemies? I can accept the red skies around Oblivion gates as being the presence of a gateway to Oblivion in Tamriel effecting the surrounding area, especially since we could also see other Oblivion influences in the immediate area surrounding Oblivion gates, but I see no logical reason why dragons would cause a change in weather.