If the rules are not clear, it's good to feel the climat of the game.

> You're walking on the streets Arkham, trying to sneak in the shadows and avoid monsters. Suddenly the wild Nightgaunt appears and the fight begins. Unfortunately you're not good in fighting and the monster grabs you, fly with you over some rooftops and throws you right to the center of the gate to the other world. You end up in the centre of cyclopic city, frightened and confused...

This seems to be quite much. I don't think, that any investigator after such situation could immediately soberly think "Meh, no problem. I have the Key of Tawil At Umr."

Moreover such usage of the Key immediately after being dropped into the gate seems not to be enough to say "I've explored this world". 

I would say, that the answer for your question is "Yes", but maybe there is someone who could provide some house rules details, that might make it unequivocal.