Day-traders tend to not call themselves investors. They call themselves traders.

Investing involves holding assets long term.

But they certainly don't call themselves gamblers. At least not any that turn it into a successful career.

You can gamble with anything, to include things that aren't worthless like food, housing, stocks, etc. You can also invest in them. But you can't invest in worthless things.

Is the day trader in your example trading their own money?

Because most traders aren't. They are bank and fund employees with KPIs, risk management oversight, massive support functions and very expensive computer programs.

Retail investors that are day trading on the other side of those professionals are absolutely gamblers though.

They are gamblers.