Just activated old iPhone and App Store is trying to make me pay for an app I've already paid for
No, Apple does not allow app creators to charge users twice for an app already purchased(in this context).
Of course the developer could create a new app with a new name, but identical content, and in that way try to get people to buy it twice. However that is not the case here.

Are you sure that you actually bought that program with that specific AppleID - are you sure you didn't buy it with a different ID?
You could check your purchasing history for your AppleID on iTunes (App Store).

Another thing to try is simply to press Download on the program, even though it doesn't display the Cloud symbol. If it downloads without requesting payment, you're good to go.