Its mostly controlled by a thermostat and if the heat pump has the inverter function.

The different brands all have different methods of creating efficiency - much like automatic vehicle transmissions.

Some modes and models may run hotter with the fan quieter, others may run cooler with the fan working more.
Mitsubishi Electric vs Kelvinator are a good example of this comparison. The kelvinator will run with a faster fan at a cooler output temperature, but Mitsubishi promote quiet operation in a few of their models and will therefore run a higher output temperature and quieter fan by design, with a slight efficiency loss.

On startup, the units make some basic assumptions about the inside/outside temperature and goal you have set.
When there is a bigger difference between inside temperature and goal, the fan will start earlier. The temperature of the air coming out will only be a few degrees warmer than inside, but the idea is to get the temperature up faster.

When the outside temperature falls below 0 degrees, the unit will often lower the output temperature and increase the fan slightly to prevent the outdoor unit from freezing up - at which point all efficiency is lost because it needs to either reverse cycle or run a heating element.
So by extracting less heat from the air outside, and therefore not outputting as much heat, the room will begin to cool. But it can speed up the fan to get the air moving around again and eliminate cooler areas in the room, rather than outputting a hotter air temp/lower fan and relying on osmosis.

So for the most part, its thermostat controlled, but inverter models can be very smart in their strive to show efficient numbers, and so the designers come up with various ways to give a balance between performance and efficiency.

When I was selling heat pumps, we used to say "Heat to 18, Cool to 23"
But in my house I just leave it set on 23