You're not wrong, but are missing an important intermediary steppe. I'm by no means a Holocaust (capital H) expert, but I've written an etymology of the word before and have come across the Greek route in readings from the Septuagint bible.

Biblically, holocaust comes from Greek holocaustos (ὁλόκαυστος), a compound word basixally meaning "a burned thing." That was the Greek translation of the Hebrew name for the sacrifice and made the jump to Latin basically unchanged when the Bible was translated into the Vulgate. The first solid use for it I can find shows up in English in the 16th century Tyndale Bible, obviously in the same capacity of sacrifice and burning as the Greek and Latin

The next most relevant uses to us are in periodicals from the interwar years in the early 20th century. By then the idea of a holocaust as a burning or sacrifice had been kicking around in English for at least 400 years and in 1925 a National Geographic article about the Greek Occupation of Smyrna and conflict with Turkey from 1919-1922. In this article its used in reference to mass slaughter in a metaphor about the flames of the conflict consuming victims.

The next two references I want to use before the coining of The Holocaust as an event are actually in connection with Nazi Germany. The first in on May 10, 1933 was a book burning in which Nazi universities burned "un-German" books and was described as Holocaust by Newsweek. The next is in February 1936 in a publication called The Palestine Press which refers to the rapid succession of German officials to Franco in the Spanish civil war as a "bloodless holocaust."

So by the time the word was used by the British News Chronicle first used the word to describe the treatment of Jews by Hitler in 1942 it had already taken on a context of general destruction in the English language. Over the course of the 1950s, the terminology was popularized by writers and historians, probably looking for a suitably graphic word to describe the event.

Edit for grammar.