Then Joseph Musashi could be half-American from his mothers side. His father Jiro was poised to take over as shuryou of the Oboro Clan but his love for Sarah Krieger may or may not be forbidden on account she is American. But their love lead to a hāfu son. Nakahara who was a "loyal" student of Jiro, turned out to be a Zeed ninja and betrays Jiro and kills him and Sarah. Nakahara then takes over as shuryou to the clan and becomes sensei to baby Joe.
Nakahara technically never lied to Joe Musashi. Joe is a half-American orphan who's parents were killed over the scandal. he just never tells him who killed them.
Actually in the game, Jiro's story is rather complicated.
He couldn't be killed by Nakahara since he's actually a master Shinobi from Kamakura Era that somehow got time travel 800 years to the present time.
After getting close to Sarah (or other woman, it's never officially revealed who Joe's mother is) and had a son (Joe), he's somehow back again to his time to lead the Oboro Clan replacing his dead (traitor) master.
Before going back he entrusted Joe to master Shinobi of the present time, Nakahara.
During the 800 years, the Oboro Clan leaders has changed many times of course until come the peaceful day in the present which makes Nakahara frustated (due lack of ninja's activity) which make him planned the Zeed thing.
Also a trivia : Jiro's Oboro Clan headquarter was on a snowy mountain, that's why the ninjas are wearing white (color of the snow).
Back to Joe, under Nakahara's guidance since he's a child, Joe became the best Shinobi of the Oboro clan which makes Nakahara proud of him so much.
Joe even found a father figure in Nakahara more than to Jiro who's already missing since he was a baby.
(Trivia : Nakahara's name is a reference to the Musashi-Nakahara Station, to show how close they really are).
That's why Shinobi's final fight is supposed to be a sad story for Joe, also paved a way to the reason for Joe to change his outfit from black to white.
May be he feels closer to his real dad now since both of them were betrayed by their own master.
Interestingly enough, Jiro's also been cloned into a Shadow Master Shinobi (which also trained by Nakahara) and took the leadership of Zeed after Nakahara died.
He also changed the Zeed name into Neo Zeed.
The final boss in Revenge of the Shinobi, the Ninja Master?
He's not the true leader of Neo Zeed.
Two years after Musashi killed him, it is revealed that he was but a mere double for the true mastermind, the Shadow Master.
That's right, the final battle of Shinobi III is between Joe and his father's clone.
Master (Joe became Master Shinobi after he beats and kills Nakahara, replacing Nakahara's position) vs Shadow Master.
Both are Nakahara's pupils.
Of course once again I'm talking about game version, movie version could be different.
Adapting time travel thing for first movie could confuse the viewers so it's making sense to be changed to make it more simple.