Translation:I pressed that round button.
There have been some inconsistencies lately with "その" being translated to "that" or "the". Technically depending on the context, either can work. To be safe, "that" is a better bet. As long as you understand the use of the "そー" words to be something/somewhere closer to the listener than to yourself, the rest is just pleasing Duo's need for chaos.
What if there are a bunch of buttons of different shapes on a board? You need to know which button. And if there are several round buttons, saying "the round button" wouldn't work. But pointing to one and saying "THAT round button" would get the point across.
As a native English speaker, I don't find "that round button" to be a strange phrase (I see you have english as a language, so I'm not sure if it's your native language or not)