I'm a theoretical physics grad student and a night school maths teacher, I have never heard this distinction. People in academia around me call them round and square brackets.

Yeah, but as an adult it depends entirely on whether you're in an industry or hobby that requires that level of bracket nuance/exponents.

Most of us are just trying to remember the basics.

The way I was taught growing up, brackets are [these]. Parenthesis are (these)

They're all brackets. Parentheses is actually the part inside the ().