All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Rape is a type of Zina (fornication or adultery), and there is no difference in proving Zina between its occurrence by mutual consent and its occurrence by coercion, but in the case of coercion, the blame is lifted from the one who was forced to have sex with someone against their wishes.
Zina is proven by evidence, which is four witnesses, or confession from the adulterer or fornicator. Zina is not proven by simply a woman claiming that a man raped her.
Ibn 'Abbas narrated that the Prophet said: “Were people to be given according to their claims, some would claim the wealth and blood of others. But the oath is upon the one who denies (the allegation).” [Muslim]
The mentioned story is true, and it was reported in the Musnad of Ahmad, Sunan Abu Dawood and others . What was mentioned apparently about the order to stone the first man, the scholars interpreted it to mean that this statement was said by the narrator due to the apparent situation, as the man was brought to the Prophet and he was busy searching about his condition, so the narrator thought that he wanted to stone him, not that he ordered to stone him in fact. This meaning was mentioned by Shams al-Haq Abadi in his commentary on Sunan Abu Dawood, quoting some scholars.
Allah knows best.