Looks good to me. You are right, the higher mah rating just means the battery will last longer as it contains more energy. Similar to replacing a car battery, the old one may have a 450 CCA rating, the new one could have a 650 CCA rating. Both are 12 volt batteries, but the higher CCA means it is a stronger battery.
The battery in the slot machine is to preserve the settings and financial data in case power is turned off. In a casino environment the machines are seldom turned off, so they could use a minimal rated battery and it would work ok. Plus the machines are often rotated off the floor before the battery has died.