It's true that
why the price levels change can make a difference. What I wrote was meant as examples of the type of relations that are there.
You're still wrong or at least extremely misleading in your characterization of how the money supply behaves.
First of all, base money is really almost entirely an implementation detail of interbank settlement and decoupled from the real economy.[1] Because of that, the focus on base money is wrong-headed and you should look at broad money instead.
Broad money grows and shrinks almost entirely due to commercial bank lending and loan repayment. The central bank has no influence on that.
It is technically correct that base money can only be changed by central bank actions, but that is a somewhat misleading statement, because the focus should be on what the chains of causality are.
We've had a perfect experiment for that in the last few years due to quantitative easing. There, central banks tried to affect the real economy by being the initial cause of a change in the money supply. Well, they successfully raised the size of base money to totally unprecedented levels, but that had virtually no effect on the broader money levels, let alone on the real economy.
All other central bank actions that change the size of base money are actually done as a reaction to demand by commercial banks, which in turn is a consequence of commercial bank lending.
Because in any case, central banks haven't even usually targeted the size of the money supply as a matter of policy. The policy has been to target a level of interbank interest rate, and let the size of the base money supply go wherever it needed to in order to accommodate the interest rate.
And when central banks tried to target the money supply via quantitative easing, which was only tried when the interest rate was already effectively at 0, the evidence is that it had virtually no effect.
There is no causal chain in practice[2] that starts with the central bank deciding to change the size of the money supply and goes into changes in price or output.
[1] It's even decoupled from the major financial markets that people are usually interested in.
[2] Of course, in theory central banks could just go nuts and start targeting the money supply in crazy ways. But even in that case, they can't just decide to cut or grow the money supply arbitrarily (because they need to find a commercial bank counterparty for what they want to do). And further, due to the decoupling of base money and broader money, the causal chain would go mostly via the effect that the base money changes have on interest rates, which is a very different story from the one that you probably wanted to tell.