Reinhardt is Richter's grandson.
Think about it: The game goes out of its way to tell us that Reinhardt is a true Belmont, yet he doesn't carry the name. Why? After Richter's actions in Symphony of the Night, bringing back Castlevania and helping to resurrect Dracula under the influence of Shaft, the Belmont clan fell out of favor with the people once again. To conceal and protect his son and family, Richter chose to give them another name, perhaps his wife's or his mother's maiden name. This son would've been Michael Gelhart Schneider, Reinhardt's father. Notice the distinct Germanic names Richter, Michael Gelhart and Reinhardt all share with one another. During Michael's lifetime, the hatred towards the Belmont line died down, leading to Reinhardt being publicly known as a Belmont even during his childhood, but the Schneider name stuck. Richter would've also strengthened his family's relation to the Church, so that if any other Belmont were ever possessed, an organization would be there to stop them, leading to Reinhardt being trained partially by the Church and his association with both Carrie Fernandez and Henry Olrey, also explaining the other Vampire hunters such as the Graves or the Baldwins. Remember, other than Reinhardt and the Wygol Villagers, we have no record of any Belmont after Richter until Julius Belmont in 1999. Perhaps the Belmont family went unnoticed throughout the years, hiding in plain sight under the Schneider name. Reinhardt was the one who bequeathed the Vampire Killer whip to the Morrises.
The last time the Vampire Killer whip was seen before it came to be in the possession of the Morris family, it was in Reinhardt's hands. The timeline fits. Reinhardt would be in his 60s by the time the original Dracula novel's events take place circa 1893. Perhaps John, having witnessed his father Quincy Morris heroically yet tragically achieving the deed of killing Dracula at the cost of his own life, sought out the remnants of the Belmont clan after discovering his own family's distant connection to them, leading to him finding the hidden Belmonts. As Reinhardt is a true Belmont as recognized by Death itself, he wouldn't have known the tragic toll the Whip would take on John's life due to him being much more removed from the Belmont line than Reinhardt.