Inflation and a rise in the cost of living are different things. Inflation means an increase in the (nominal) price level, whereas the cost of living is measured in real prices, specifically real wages.
Inflation is an increase in the money supply, which is why it's called inflation - the money supply inflates. I know a lot of government economists like to define it as rising prices because then there is no word for increasing the money supply which is very convenient for them and the governments they work for.
No, it's not just government economists. It's the standard definition in economics. But that's beside the point... the BIG mistake you're making is confusing an increase in nominal prices (inflation, or whatever you want to call it) with an increase in real prices (a rise in the cost of living).
