If you open your Nest App up and set the mode to Away, it will set both Nest Presence and the tile in Nest Thermostat to Away.
For NST Manager Automations / Modes, if you create/set this up to syncronize between ST and your Nest Thermostat, it uses the ST Modes (Home, Away, Night) to send to your Nest Thermostat so your Nest environment goes to Away when ST goes to Away and vice versa (this is how I use Nest and have Phone Location turned off in Nest).
So if you are using WebCoRE (Pistons) or Routines that arms/disarms ST (SHM armed status and changes the Mode, you don’t have to do anything else if you have the Modes Automation setup in NST Manager. This will automatically change your Nest Thermostat in and out of Home and Away based on the Mode in ST.
You don’t need to utilize the Nest Presence or Thermostat devices to do anything else with this setup.
You can also setup Thermostat schedules directly in NST Manager which you can see some of what is being discussed above based on ST mode, time, etc…
Now perhaps you don’t want to use the NST Manager schedules and maybe have some sort of unique or more complex set of requirements (a lot of ifs / ands or if/ors) where you want to define certain things to happen with the Thermostat, then you could leverage the Nest Presence sensor as the device that you base your qualifications on (Present / Away). As for the Tile in the Thermostat that shows Away or Home - direct reflection of Nest), that is used as a visual representation of the Status of Nest only. Not used in Automations.
FYI - The Nest Presence represents Present (which is Home) and Away which is away. The Present is misleading.
I would leverage the Mode / Thermostat Automations (it’s pretty robust what you can schedule things for or base things off of) as much as you can. If you find that your use case cannot be satisfied using the internal Automations, then and only then would I venture off in creating Routines or Pistons that satisfy those requirements.
Hopefully that answers some of your questions and clears things up a bit.
Some screenshots for the Modes Automations if you haven’t looked at it in NST Manager yet: