We have a 120ft well with (of course) a submersed pump. It also has a manual faucet coming out of the well for what I suspect is for using for gardening and such so as to not use the filtered water inside the house. I tried using it for the first time today and as soon as I lifted the handle up it turned on the pump (I could hear it) and a ton of water comes out. Thing is though, the pressure switch inside the house cut off and stopped working and had to figure out that I needed to manually had to turn up the side arm to fill the pressure tank back up and things started working again. Why can I not use the manual faucet in the pump house outside without messing up the system inside?

Is there a checkvalve between the faucet and the pressure tank?

The only reason the pump would cut on would be because the switch activated. It could be as simple as the pressure was, as luck would have it, already near the cuton pressure. When you opened the faucet, the pressure dropped the extra two or three pounds needed to activate the switch, which is why the pump cut on as soon as you opened the faucet.

The switch has a safety feature which disables it once the pressure drops too far below the cuton point. It just "assumes" the well is dry and the pump needs to be cut off. That's why you had to reset it. Cutting on the faucet must have allowed more water out than the pump could keep up with and the pressure dropped too far.

Just guessing.