Why can't it be both monetary and non-monetary? Say it's a vector, one element per CPI category. Throw housing in for good measure. The direction of this vector can change due to non-monetary stuff like Russia and oil. But if all of the categories, especially those without clear non-monetary drivers, rise, then it's also monetary. So maybe X = p_monetary + Q_nonmonetaty where p is a scalar and Q is a vector. I think…
So when you introduce money through debt, what you get is mostly the third group who takes out debt. If they have the money and don't actually spend it, you just get a lower velocity, you don't get any inflation. When assets are increasing faster than consumption items, of course they invest in assets, and you get stocks and homes and monkey jpeg reciepts going up.
That is, until a recession is coming around. When a recession is incoming, money managers look at history and find the best recession-proof investments. And it turns out some of those items are in the consumption basket. And it turns out widely inflated asset prices are exactly what you need to get out of.
What happens when you buy oils futures contracts 2 years out? Some bank will work out a arbitrage opportunity, hedge that contract, some other bank will hedge them, and within a few days the value of oil TODAY goes up. That's inflation.
And so you can say that expectations of interest rising causes recession fears, and those recession fears cause inflation. If the money supply drops, or is expected to drop, or we think that the likelyhood of debts getting margin-called is going to increase, you will see inflation.
But you can only see that inflation, as Friedman rightly pointed out, if the reservoirs are full. Wealthy people store possible inflation in their reservoirs. If as the Fed you completely ignore the possibility that the dam can release all of that water out into the rivers, you're always going to be surprised when it happens.