All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The name of Loot (Lot) is not derived from "liwaat" and "looti" as you mentioned, and linguistically, "liwaat" does not mean homosexuality. The origin of the word in Arabic means sticking.

Therefore, the name of Loot who was a prophet of Allaah is not derived from the word for homosexuality as you understood.

Ibn Manthoor said, "Loot was a prophet whom Allaah sent to his people but they denied him and they invented homosexuality, and then the people derived from his name the name ascribed to whoever commits the act of his people.”

Hence, it is people who derived the act of the people of Loot from his name and they described with this name a person who commits the act of prophet Loot's people (homosexuality).

Allaah Knows best.