That's not quite right, because if you only reduce demand you could also reduce supply and prices would be unchanged. Higher interest rates reduces money supply (which can induce a recession) but does not necessarily reduce the supply of goods. Less money chasing the same amount of goods (ideally) causes lower prices.

I don't think you are wrong intuitively. I am just trying to be a little more specific because get very vague about monetary policy and it leads to some bad assumptions.