If you are being observed then you can't hide while they are watching you hide! This is not the same as the very act of looking around for possible hidden foes meaning that you automatically spot them!
It's not
necessarily the same. It could be the same. The former is one subset of the situations described by the latter. Whether the latter is true is subject to interpretation by the DM, and you could make the case either way. I'm taking the interpretation that it is true, because it makes the game run faster and is more player-friendly, and because it makes more intuitive sense to me.
If you have two people in a 10' x 10' room, you just
can't hide. It's flat-out impossible. Even if you want to wait in ambush of the other person entering the room, there's simply nothing to hide behind. Once they enter the room, you've been spotted... unless they aren't expecting you (so they aren't paying attention all around them, so you might be able to sneak behind them immediately as they enter). But once they know you're there, there's no point in trying to hide, because they can definitely see you. You can
always see something in front of you if there's nothing in the way, and it doesn't take an action or anything to change the direction you're looking.
Now replace that 10' x 10' room with a 20' x 20' forest clearing, and the same logic holds. If you're
in the clearing, and there's nothing to hide behind, then they can see you. They just can. You're not
behind anything, so there's nothing to
stop them from seeing you. (Unless you want to bring up an exception for camouflage for something.)
Now add some waist-high tree stumps that you can crouch behind. When you're down behind one, and it provides cover such that you are not visible, then the other person
can't see you... but they might be able to hear you and so you get a Stealth check to hide. If you pop up, though, then you no longer have cover and you are immediately visible to them because there's nothing stopping them from seeing what's right in front of them. If you want to sneak around behind them and
then pop up, where you think they won't be looking, then I think there are advanced rules for facing or whatever in the DMG. The basic rules in the PHB don't deal with facing in combat, and leaves it up to DM discretion outside of combat.