Using mathematical nomeclature outside of the well-defined realm of mathematics often yields interesting problems. What for example do you mean by "proof" if you're not talking about logical deduction of a theorem from a set of axioms?
So when I talk about optimization I usually think about some more or less well defined problem, like "Given a set of factories and clients, how do I transport goods to clients using the minimal amount of money. I only have a limited amount of trucks and drivers, so there are some complicated constraints". The tricky part is judging which factors are important to my real world problem and modelling them using well-defined mathematical objects. This transition from real world to mathworld is usually hard and requires several attempts. The result of the process is the optimization problem that we want to solve using mathematical methods. But note that it's only our approximation of the real world and we left out lots of things that we deem irrelevant!
Most real world problems are of course difficult and any model that can give reasonable results is at least NP-hard to solve optimally. So we use heuristics. I still think of this as an optimization process, since I do improve something until I reach some kind of solution. It might not be the global optimum, or even a local optimum, but it's the best I can do. Since my original formalization of the problem already left out lots of things that might or might not have influence on the solution, this additional layer of suboptimality does not matter too much to me.