Whitney was an emotional girl, still getting used to her role, but ultimately realising a defeat was final and as Gym Leader, there would be times when she was defeated, as her role was to test the trainers attempting the Gym challenge to prove them worthy.
Clair was the crybaby, so used to defeating trainers summarily that when she was faced with a trainer able to best her, she went into full denial mode and started trying to engineer a way to refuse the badge to you for failing ALL of her gym challenge, by adding the extra stipulations after the battle.