I've never actually seen anyone use "inflation" to refer to cost of living. Most economists use the term to talk about price increases of a subset of goods, but that isn't directly measuring cost of living.
Most people are not economists. All they know is they're spending more for less every month, and they don't like it.
The money supply idea has always been Austerian crackpot nonsense, intended to dissuade governments from investing in public services of all kinds because that's socialism, and we certainly don't want any of that.
In reality price rises are mostly driven by corporate profiteering, and sometimes - as in the oil crises of the 70s - by supply shocks.
