The 2nd degree manslaughter requires negligence. NY law didn't allow him to exit the house with the gun, because the punks were only threatening severe harm. The threat they posed to to his kid was rather clear, and so are the possibilities of what they would have donethen, or later. The threats in this case were real, and not idle BS. White didn't really think this was a lynch mob. The old stories just popped up to enforce the anger that night and probably triggered the, "I'm not going to put up with this shit" response. After all the threats were real, and he didn't know who he was dealing with, why, or if it would not end with his kid beaten to death somewhere, some other day, or if his house would be burnt down.
I don't think White thought much about being attacked otherwise for a long time. Being woke up in the middle of the night by a mob outside demanding to have access to your kid, so they can beat him, as they see fit too can generate some strong reactions.
The kid Mr. White shot went to school with Mr. White's son. The young men knew each other by name and by face and had spoken on the phone. Mr. White could have called the police and identified those who were arguing with his son and let the police deal with it. Instead Mr. White chose to handle things his own way which culminated in his firing a pistol into an unarmed young man's face from point-blank range. Mr. White screwed up royally here.