No, that’s not what I mean.
Show me any video of what you just described, however, remaining in view for 8 to 10 minutes, and I’ll grant you your point.
I think I'll stick with Gil Leyvas' estimate of 3 minutes maximum.
But this one does show a lot of similarity even though the second stage is powered flight. In the first stage there is a continuous plume, in the second stage there is a plume that doesn't persist long and very briefly after that there is a small tail plume barely visible even though the third stage is a solid rocket motor also.
Delta 2 STSS Demo Rocket Launch - SpaceflightNews.net
This video doesn't show the contrail after the vehicle is gone for 8 - 10 minutes, as Leyvas put it, but it does show a few similar things.
It takes almost four minutes to fly out of sight. It begins with a very thick plume which diminishes to near invisibility even though it is under power the entire time. The audio picks up no sonic boom even though the cameraman is almost directly under it. The flame of the rocket motor is obscured much of the time. Its flight describes an arc even though it is being launched into orbit.