Intuition for effect of interest rate changes in New Keynesian model
Yes the 'standard story' you reference is the correct intuition here. Lower nominal interest rate encourage consumption which puts pressure on both output and prices.
Household can actually save and borrow in the model. However, because it is a representative household model in equilibrium there would not be any borrowing - the interest rate should just adjust in order to make agent's indifferent between consuming today and in the future - but the intertemporal consumption choice still depends on $i$ with higher $i$ incentivizing  future consumption at the expense of the present one.
Also, even if in the description provided in question this is not shown the Euler equation has its micro foundation in model where households are allowed to save and borrow - otherwise the interest rate would not be part of the Euler equation in the first place, and I am not sure how it would even be modeled if there would be no link between present and future. For example, see the treatments of IS-LM models in lets say Woodford (2003). Interest and Prices. You will see that in dynamic models the Euler equation is typically derived from model of household behavior where households have access to saving/borrowing.