Thanks, Diana, I see what you mean now, and that clarifies what I miss about the old phone: the notification didn't go away when I looked at the text or calendar or whatever; it stayed in the notification drop-down until I cleared my notifications. In other words, clearing notifications was a separate operation. For instance, I might read a text but want to wait until later to respond, so I'd leave the notification in place as a sort of to-do list that I could access with a single swipe of the screen.
I guess the only way to do that on the iPhone would be to create another notification...?
I noticed that the text preferences screen lets me set repeated notifications for texts, but that would be annoying for texts I don't need to be reminded of, and doesn't solve the lack of 'persistence' for all the other kinds of notifications I might want.
Anyway, I appreciate all your help with my niggling little complaint, Diana. I'll just have to figure out how to make this system work with my spacey brain, even if it means sticking post-its to the back of the phone. Talk about first world problems!