Most simple macro models have just one (nominal) interest rate. I want to complicate it, just a little, by talking about two (nominal) interest rates:
1. There is the rate of interest you get paid if you hold money. Call it Rm.
2. There is the rate of interest you get paid if you lend money. Call it Rb.
For a paper money, Rm is nearly always fixed at 0%, simply because of the practical difficulties of paying interest on paper currency. But for an electronic medium of exchange, there are no practical difficulties, and Rm can be positive or negative. For example, chequing accounts sometimes pay positive interest (or reduce your fees if you hold a large balance). And commercial banks may get paid interest on the electronic money they hold in their accounts at the central bank. But Rb is generally not the same as Rm. Rb is nearly always greater than Rm (otherwise people would simply hold money rather than lending it). Except for the constraint that Rb cannot be less than Rm, the two interest rates don't have to move together.
Now for my simple question:
If you believe that "Central banks control monetary policy by setting the rate of interest" which of those two interest rates do you have in mind? Is it Rm, or Rb?
Old Keynesian ISLM models assume that the rate of interest is the opportunity cost of holding money, so they implicitly assume Rm=0%, so "the interest rate" means Rb.
But in simple New Keynesian models, all money is electronic money and the central bank sets the rate of interest on electronic money, so "the interest rate" means Rm.
That's an important conceptual difference between those two "Keynesian" models.
We can imagine a world where all central bank money is electronic money, and the central bank can alter both the quantity of money and the interest rate paid on that money, and can make Rm and Rb move by different amounts, or even in different directions, if it wants to.
To my monetarist mind, an increase in Rm increases the demand for money, and that causes an excess demand for money, just like a reduction in the supply of money causes an excess demand for money. An excess demand for money, or an excess supply of money, has macroeconomic consequences. Any change in Rb is just one symptom of those macroeconomic consequences. We would get roughly the same macroeconomic consequences even if Rb was fixed by law, or if lending money at interest was tabu.