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.
People view risks differently, some stake a huge amount of money on a low value odd and to them that's the best way to gamble responsibly and manage risks, while others stake low and aim for big odds...for me I would choose the second option because in gambling Small odds don't guarantee wins, sometimes it's even the big odds that are successful...the way to manage risks is to gamble with little amounts