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Maimonides referred to Muhammad as a false prophet and an insane man. In his Epistle to Yemen he wrote "After Jesus arose the Madman who emulated his precursor Jesus, since he paved the way for him. But he added the further objective of procuring rule and submission [talb al-mulk; pursuit of sovereignty] and he invented what was well known Islam. (source wiki)
Did Maimonides really refer to Muhammad as a false prophet and a madman?
Maimonides did call him a madman. See the translation of his Epistle to Yemen here (3rd paragraph) or here on Sefaria (search for "madman").
To summarize the historical background (brought here)
About the year 1168, the Jews of Yemen were confronted with a three-pronged agonizing problem. A fanatical Moslem cleric became the ruler of this distant, primitive South Arabian land and decreed that his Jewish subjects convert to Islam under the threat of harsh punishment and suffering.
Their agony was compounded by a Jewish apostate who embraced Mohammedanism. To demonstrate his zeal for his newly adopted faith, he began preaching to the Jewish communities that Mohammed was a divinely sent prophet alluded to in the Bible and that Islam was a new, divinely revealed religion superseding Judaism. Hence, the apostate argued, the Jews should yield to the ruler's demand and embrace Mohammedanism. [...] Rabbi Jacob al-Fayumi turned to Rambam for advice and counsel.
Rabbi Moshe ben Maimon [Maimonides] addressed a letter to this sage, and through him to the entire Jewish population of Yemen. He states that the root of all anti-Semitism throughout the ages is envy of the Jews being the Chosen People and the recipients of the G‑d given Torah. Unable to do battle with the Almighty Himself, the haters turn their jealous rage toward His people.
Throughout the ages this has taken three forms [...] [3.] the false claims of new religions - Christianity and Islam - that Judaism is no longer valid and Jews must be forced to accept the new revelation. He consoled them by telling them that the Jews are a unique and indestructible nation; that all the past and present sufferings and persecutions were foretold by the Prophets, and just as in the past the nations had failed to annihilate the Jewish people or destroy the Jewish religion, so will the present persecution fail, and peace and tranquility will return to the community.
He contemptuously dismissed and disproved the assertion that Judaism has been supplanted by Islam and showed that the claim that Mohammed is alluded to in the Bible is based upon nonsensical interpretations recognized as such even by the Moslems themselves.
Rambam advised that the self-proclaimed Messiah is nothing but an impostor and no doubt a madman. He urged them to remain firm in the belief that G‑d will send the true Moshiach to redeem the Jewish people from suffering in exile at the proper time.
The epistle accomplished its purpose - the Yemenite Jews remained faithful to their religion in the face of their bitter suffering. Rabbi Moshe ben Maimon used his influence at the court of Saladin in Egypt to intervene in their behalf, and the persecution came to an end.
As noted in your original source, in his compendium of Jewish Law (Mishne Torah Melachim 11:4), Maimonides wrote that Jesus and Muhamad served to prepare the way for the true Messiah by turning people to monotheists.
Ultimately, all the deeds of Jesus of Nazareth and that Ishmaelite who arose after him will only serve to prepare the way for Mashiach's coming and the improvement of the entire world, motivating the nations to serve God together
