Obviously it works well enough for mainstream economics, but why does no one explicitly qualify inflation as "price inflation", "demand inflation", or "monetary inflation"?
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.
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.
and the original interview is informative too. (tl;dr: Great Inflation)
It's actually worse because economists themselves use the same word to describe the two different concepts.
I agree with Symmetry above that the simplest 90% solution is just use "inflation" for general rise in price levels regardless of cause, and "monetary inflation" for money supply expansion.