Prerequisite: intermediate macro.
I reckon some people might be getting those two things muddled. The difference matters. Actually, the difference is the only thing that matters. It's the spread between those two interest rates, not the levels of those two interest rates, that matters for Aggregate Demand.
Simplify massively. There are two rates of interest. There is the rate of interest you get paid for holding money. Call it Rm. And there is the rate of interest you pay/get paid for borrowing/lending money. Call it Rb. Those are two conceptually very different interest rates. Rm is the rate of interest you get paid for the media of exchange in your pocket or chequing account. Rb is the rate of interest you get paid for the bonds or non-media of exchange IOUs you own.
[Update: I have a new post that adds a picture to this one.]
For example, take the very simplest textbook ISLM model. There appears to be only one rate of interest in that model, the one on the vertical axis. There are in fact two rates of interest in that model. The one on the vertical axis is Rb. The other rate of interest, Rm, is implicitly or explicitly assumed to be fixed at 0%. We know that because, as all the textbooks say, "the rate of interest is the opportunity cost of holding money". That statement only makes sense if we assume Rm=0%. Because the opportunity cost of holding money, rather than lending it, is Rb-Rm. So the rate of interest on the vertical axis must be Rb.
If we assumed that Rm were not fixed exogenously at 0%, and if we instead assumed that Rm followed Rb up and down, so that the spread between them Rb-Rm were a constant, then the LM curve would be vertical for a given Ms. Because there would be only one level of Y at which Md=Ms, if Md=L(Y,Rb-Rm). Where LY>0 and LR<0. Which would give us a very different version of the textbook ISLM model. Because shifts in the IS curve would have no effect on Aggregate Demand.
Start in textbook ISLM equilibrium. Now suppose there is an exogenous increase in expected inflation. It rises from 0% to 2%. We stick a 2% vertical wedge between the IS and LM curves, with the top of the wedge on the LM curve, and the bottom of the wedge on the IS curve. Because saving and investment depend on the real interest rate, and the demand for money depends on the nominal interest rate, and nominal - real = inflation. That 2% expected inflation wedge shifts the Aggregate Demand curve to the right.
But another way to say exactly the same thing is to say that when expected inflation rises from 0% to 2%, the real rate of interest on holding money drops from 0% to minus 2%. And that fall in the real rate of interest on holding money is what shifts the AD curve to the right.
You could get exactly the same rightward shift in the AD curve, holding expected inflation constant, by assuming the nominal interest on holding money drops from 0% to minus 2%.
Aggregate Demand does not depend on the level of interest rates. Aggregate Demand depends on the spread between Rb and Rm. If that Rb-Rm spread is constant, so the opportunity cost of holding money is constant, the LM curve is vertical for a given M, and shifts in the IS curve have no effect whatsoever on AD. And shifts in the LM curve, caused by changing M, cause the AD curve to shift by exactly the same amount, regardless of whether the IS curve slopes down, up, or sideways. It simply does not matter what happens to the level of interest rates, if the spread between Rb and Rm is held constant.
Monetary policy works by changing Ms, and also by changing Rm to change the Rb-Rm spread.
[The interest rate in New Keynesian models is actually Rm. There are no bonds, so there is no Rb. Everyone has a chequing account at the central bank, which can have a positive or negative balance, but the net aggregate balance is fixed at zero, and if agents are identical each agent chooses to hold a zero balance in equilibrium.]