Confusingly, while rises in the general price level are called inflation and that's what pretty much everybody means when they talk about inflation a proper term for an increase in the money supply is also inflation. I would strongly recommend people not use the term "inflation" that way unless you're careful to distinguish the sense in which you're using it, since you'll usually just end up confusing everybody otherwise.
An increase in the volume of a helium-filled balloon is also "inflation." As an economist, I can assure you that I mean "price inflation" when I say "inflation" without a qualifier. So does everyone I talk to. It doesn't need to be confusing.
There's clearly a need for a word that means, "increase in the price level" and that word is "inflation." Your nitpick almost reads like a criticism of the word "fever": it doesn't distinguish the causes of the fever but lumps them together.
Economists have been studying inflation long before they really understood its causes and there are surely more refinements that are left, so it's vital to be able to talk about the observed phenomenon in a cause-agnostic way. Just like it's vital that doctors can talk about "fevers" without having to know about the exact underlying cause.
Economists do distinguish and discuss the causes of inflation. Search for "demand pull inflation", "cost push inflation" or "imported inflation". You won't see "monetary inflation" that much in modern economics discussions because it simply isn't that relevant empirically for understanding the development of the price level.
He was wrong. Certainly even the bit of truth that is left in his statement is mostly misleading. Inflation is always better understood by looking at prices: What is it that leads businesses to change their prices? Why do they raise prices, and why do they get away with it?
For a bit of back story: What this quote of Friedman is intended to convey is the incorrect idea that there is always a causal link from an increase of the money supply to a rise in the price level. The rise in the price level is explained by a prior increase of the money supply, and this increase of the money supply is supposed to be the prime cause.
This is wrong on several counts: (1) there can be a causal link from the price level to an increase of the money supply, because money is endogenous; (2) as neither the real size of the economy nor the velocity of money (Q and V in MV = PQ) are constant, the price level and the money supply can move independently from each other; (3) in practice, growth in the money supply and rises in the price level are both caused together by other causes (such as rising cost of imports or institutionalized wage increases).
The most important thing to understand is that our monetary system is endogenous: How much money is out there is decided by banks and debtors. Every time a business goes to a bank and takes out a loan, the money supply grows. Every time a business pays a loan back, the money supply shrinks.
This explains why the causality tends to go the other way from what monetarists believe: When wages rise or the cost of inputs of production rises, businesses take out larger loans to cover their upcoming production, hence the money supply grows as a consequence of inflation.
Friedman and other monetarists are very much working from a gold standard and hence exogenous money mind set, so it is no surprise that they get this wrong: they are starting off from incorrect assumptions about how our monetary system works.
That is not to say that it cannot go the other direction. It can go both ways. Which is why what one should really look at is how prices are set in the economy. Since most prices are administered somehow, usually controlled by longer term contracts, you need to understand how the negotiation of those contracts works.
This has always irritated me. Normal people and economists use the same word to describe two different albeit somewhat related things. If you include debt (and I think you have to), we had some pretty severe deflation starting 2008. But the prices people pay for the things they buy went up during that time period, something that can happen while you're in a deflationary economy.
Falling interest rates usually result in increases in circulating money supply. Supply/demand definitely has an effect, but then so do increased living standards (more taxes, more pay to those who stock coke machines). Coke in Switzerland is not expensive because of inflation. With increased living standards, lower end goods necessarily become more expensive while higher end goods become cheaper (or at least don't rise so much).