The h_t in my example is a very specific model for illustrative purposes where labor is fixed. Obviously that is not true in reality.
DSGE models in the end almost always only model hours. But if you start with a neoclassical growth model, then you usually start with hours per person multiplied by the number of people and detrend from there.
Which population concept you use for per capita/per worker values again depends on the model at hand. Total population is usually appropriate if the labor force participation rate has not fluctuated too much. Otherwise, you would like a measure of workers. See e.g. the online appendix to Smets/Wouters.