So, I was right.
My friend was trying to convince me the underdogs bonus is only given when you kill anenemy, as you said, the bonus is for surviving against a stronger wanband than yours.
In the book it's not clear enough
"When a warband fights against an enemy warband with a higher rating, its warriors earn extra Experience points as shown on the table below. The higher the opposing warband’s rating the more points the underdog earns."
Even in the French version my friend is using it's not clear enough ( for him)
In anyways, Thanks and I will get my bonus xp asap