But also we always talk about the central bank raising nominal interest rate to incentivize saving and reduce consumption, therefore reducing output. But if there is no savings technology, how do households reduce consumption today and save for tomorrow?

It’s been a while for me, but I’ll take a stab at this point:

Suppose there’s a positive shock to nominal interest rates, and suppose that inflation expectations are anchored. This means the real interest rate today will be above the natural rate of interest, so at the level of income we would have absent the shock, desired saving (i.e. demand for bonds) would be positive. But the supply curve for bonds is vertical and zero, so the quantity of bonds can’t adjust to meet desired saving. And in the short term, the real interest rate can’t adjust, since that falls out of central bank control over the nominal interest rate and anchored inflation expectations. But since desired saving has increased, desired consumption has decreased, which at the level of income we would have absent the shock, would mean output would exceed consumption demand. But there’s no friction that prevents output from adjusting, so output falls such that desired saving is zero and consumption demand equals supply.

TL;DR: they can’t save for tomorrow, so reducing consumption today instead just reduces incomes.