The author himself lacks to provide more backstory for Zabuza.
From what we can tell is, that exam is intended for selecting the potential successors for Seven Swordsmen of Kirigakure, and was in practice even before Fourth Mizukage.
And the class of Zabuza is last for having such practice, as he killed the entire class, not just his opponent.
As for the reason or his motive, we will have to look at his character during Land of Waves arc. His working under Gato as mercenary is to get enough funds for starting another coup at Kirigakure.
And his reason for starting the first one? His believe is shinobi should the ruling class of the land, and during his time at Kirigakure, had disagreement with fourth Mizukage(Not sure if already under Obito's control). And the failure is said to be betrayal from one of his team.
And his interaction through the Part 1 arc defined him as cruel and arrogant one. And see his subordinates as expendable, until things were changed with Haku.
So for his motivation in killing his entire class in graduation rite, it's not like how being shinobi is for Itachi or Yashamaru(Gaara's Uncle).
We would also want to recheck on meaning of Shinobi or Ninja, as there are many portrayal of aspects of being shinobi. Some could be nobel as Hashirama and protagonist Naruto. Or could be cynical, like most adversaries of the series. Also there are pure ideological like in some of Akatsuki, Madara, Obito or Pain, who wanted to end all the suffering.
As for Zabuza, his cruelty surfaced even during the academy class, and also the environment of Kirigakure at that time was a contributing factor, survival of fittest, and unrestrained violence.