The rich have better ability to control their gambling orders, and their emotions. Because as you discussed, they know that they only use small part of their balance to bet, so, if they lose with that bet, they won't lose all, hence help them to better control their emotions. For the poor, they mostly gamble to earn money, not for fun, for entertainment like the rich, so when they lose their first bet, their emotions change dramatically. They might go crazy, and then losing more money with their next gambling orders.
Rich or poor, it does not matter. If you get addicted to gambling, you have the tendency to lose control. Why else do you see rich people squandering their wealth to casinos?
The only difference I see is that rich folks got more money to spend in their gambling. They have a thick buffer before their wealth run dry compared to the poor people. But the end is the same if they don't get their acts together.
You need to realise that most people who get addicted to gambling are not rich, this is why it becomes an addiction so easy for them because they have a "way out of being poor" and become addicted.
The rich don't generally have this issue. They basically gamble for fun and entertainment. If they lose, they accept it, if they win, then its great. However they don't do it to earn an income or to "Be a billionaire instead of a millionaire".
The poor gamble and win $500 easily, which could be a weeks of full labour, then they think how about I gamble more and make $5000. However normally what happens is they lose the $500 they won, and then they start losing their own money and go broke. Some even take out credit cards and gamble with money that's not theirs.