Inflation is the measure of the price of goods
> It's the result of funding the deficit by running the printing press
Increasing the money supply indirectly increases the price of goods. It matters in the long run, but the price of goods can change in the short-term based on supply. The oil crisis of the 1970s caused oil prices to increase because oil supply was limited. This would have caused inflation even without an increased money supply.
> creating debt backed by nothing. ... fiat money systems.
This is more of a tangent, but the money supply can also shrink in fiat systems. Fiat currency can be shrunk via taxes. Debts can be defaulted. Fiat systems aren't inherently inflationary; but they target a low level of inflation because they prefer it to deflation.