There is a clean word that defines person that is invited: an invitee.
However, I can’t seem to find a straight definition of either terms that would define a person who invites the invitee.
Is it “Inviter”, “Invitor”? Is there some other term that is more appropriate?
You could use Host or Hostess for its feminine counterpart.
someone who invites people to a meal or party, or to stay in their home.
However, as pointed out in the comments, it will not be suited in all contexts.