“Inflation = increase in money supply” is a definition used by the pro-fixed-money-supply crowd.
However, the vast majority of the world uses a much more useful definition, which is an overall increase in prices (and sometimes, an overall increase in prices that’s not due to an external force such as oil prices rising due to supply restrictions).
The money supply needs to grow and shrink with the size of the economy. If it’s very far out of whack from that, it rewards behavior that’s not economically productive. (That in turn tends to cause the economy to shrink, which exacerbates inflation, but tends to stall deflation. Thus, we don’t see a lot of long periods of deflation.)
Very few countries use the US dollar as their currency or as a backing to their own. Panama used the US dollar when I was there; perhaps it’s changed, but that’s fairly unusual (and due to the Panama Canal, which was US possession at the time.) New Zealand sets a rate of conversion between $US and $NZ, so the NZ dollar is pegged to the US, but not backed by it. Many countries (e.g., China) buy large amounts of US Treasury instruments as an investment, but not to “back their currencies”. Most currencies are fiat currencies, as mentioned above. They have no intrinsic values; each is backed up only by the stability of the issuing government, its fiscal policy, and its national economy.