I think responsible risk is the same as only using money that we can afford to lose.
If we can only afford to lose $200 a month then only use that, there is still a risk of losing it all but if that happens it will not affect anything, because that is the amount we can afford to lose.
But it could also be for other gamblers responsible risk is how to use the money to gamble, meaning with a small capital then will play casino games with low volatility, or when betting on sports betting set a small betting percentage to suit the budget.
The latter though is what a strategy is and not more of a calculated risk. But I do agree with your first definition that we should "gamble the money that we can afford to lose". And so even if we lose then nothing will be affected. We can wake up in the morning and not being traumatized by the money that we have lost the night before. And then maybe stop for a while, who knows.
With that, a gambler that uses calculated risk is more of a thinking gambler, a intelligent one. He knows what is at stakes and could make his bold decision to bet big or not stay away. This is in contrary of what we call YOLO gamblers.