The problem often lies with how different people describe things. Many people still get integrated/internal(semi integrated) mixed up. Internal ones have cups that press into the frame & the bearings then drop into them, so the entire headset is inside the headtube.
Integrated ones have a chamfer inside the headtube itself for the bearing to sit directly on, which can be cut at either 36 or 45 degrees, depending on the design.
Hidden headset I don't know to be honest, sounds like someone's given one of the above yet a different name!