In the Batman nniverse, it's safe to say the incident in Crime Alley is what turned Bruce Wayne into the Caped Crusader. Seeing his parents gunned down in cold blood revealed the dark side of Gotham and the world in general, which is why so many cartoons, comics, films and such use this as the catalyst for change. However, while it initially felt like Christopher Nolan used the same strategy in Batman Begins, it wasn't the Wayne murders that created Christian Bale's Dark Knight -- it was Carmine Falcone.
For years, Bruce held the anger within, which is why he wanted to kill Joe Chill. Unfortunately, Falcone's goons got to him first so Chill couldn't spill secrets about the mob boss. It dismayed Bruce, who harbored guilt and sadness, acting out to the point Rachel slapped him for having a gun. She didn't see a man or someone virtuous -- she saw a spoiled brat who tried to sully his parents' legacy.
Enraged, Bruce would head to a club at the docks, eager to confront Falcone, but this is where he finally realized how low he had sunk. Falcone's thugs forcefully sat him down as the gangster highlighted all the cops, judges and politicians on his payroll, making it clear he could shoot Bruce dead right there and no one would blink an eye. He also called out Bruce's rage, indicating he was afraid and not thinking things through because he still had Rachel and Alfred to lose.
Falcone would then have his goons beat Bruce and throw him out, admitting he liked his spirit but thought his actions were funny. He even joked about how Chill said Thomas "begged like a dog" when he killed the couple. The despair in this moment, however, brought a new purpose to Bruce, with him realizing the world was bigger than him and a larger symbol was needed -- something immortal that could operate outside the jurisdiction of the law.
With that being the case, Bruce had to get rid of his fear and transform it into the fear of others. It's why he left, shed his coat and left his belongings to a vagrant. He then faked his death and got on that boat to become a new person, ultimately training with Ra's al Ghul. That means that, as formative as his parents' deaths were, it was Falcone's words that made Bruce return with his new mission.