when the aspect ratio is not mentioned it's assumed that it's 82, 145/80/15 is nearly the same but not exactly the same (although with production tolerances and all it ends up being the same)
145/80-15 is the other way of marking a 145-15 tire. It is the same size. Like you, I was also told that in reality it means that the profile is 82%.
Why is it marked as 80 (or left unmarked) then? I don't know. Maybe to keep people confused

But there are no two different sizes - 145-15 and 145/80-15. They are the same thing. At least I think so.