You've answered your question with the very first word, 'supposedly'. The autopsy, conducted by surgeon Dmitry Kosorotov, reveals three important facts: he was shot three times, he was dead by the time he was thrown into the river, and likely never poisoned to begin with. Though this autopsy might be considered flawed by today's standards, the results are the best hard facts that we have, as the entire murder took place behind closed doors.

The account of Felix Yusupov, a conspirator and the owner of the house in which the murder allegedly took place, is quite shaky. His account of events is where the claim that he was poisoned originates, as well as some more supernatural details, such as a Rasputin who, after being apparently shot dead, comes back to life to attack Yusupov himself. How true this account of events might be, and why Yusupov would include such fantastical elements in his story, will likely be a mystery forever.

It's important to remember that Rasputin was and is a man of legend; a story people believe in. Even at the time of his historical death, myths about the event were everywhere. The ambassador of France notably reported that Rasputin had his penis completely severed from his body, although the autopsy was able to produce some very convincing evidence to the contrary. The facts of his death are likely much less glorious: shot a few times and then hastily disposed of in the nearby river. We like to believe these things aren't true because it's fun, even if the myth doesn't necessarily line up with the facts. The Boney M song is far from historically accurate, but it's exciting and it's fun, which is why we hold on to such stories in the first place.

Tl;dr, “supposedly” doesn't mean “definitely”, and sometimes it can really just mean the opposite. We'll never know what happened to Rasputin for sure, but the world prefers to believe a much more exciting story instead, even if it likely isn't true.