Are you saying this root is akin to the English word "messiah" and was translated as "anointed" in English versions of the Hebrew bible?
Yes: the English word "messiah" is not native to English, it is simply a borrowing of Hebrew
moshiach from this root M-SH-CH "to rub with oil" which also shows up in English as the word "massage" (another borrowing, this time through Arabic).
Is one who undergoes this ritual considered a Messiah?
The Hebrew
moshiach came to take on a lot of heavier meanings than just someone who had been installed in office with an oil ritual.
What are some other big names that underwent this ritual?
Every king (David, Solomon, etc.) and every priest (Aaron, Zadok, etc.) underwent this ritual to assume office. The same was true in a lot of other cultures as well: in India, the sacred oil is called
ghee from the same root as English
grease or Greek
christos (it has that "weak liquid" L/R that I have mentioned before, here vanishing in the Sanskrit though preserved in English and Greek; Avestan should have a similar word, but I don't know offhand what it is); but aside from use in ritually anointing a new king or priest, it is also mentioned in the "Laws of Manu" as a nice thing for a wife to do for her husband after he has a bath (that is, in India too it could just be a "massage" rather than a "Messiah").
Did Jesus undergo this ritual?
The prophet who came before the Messiah (in the capital-M sense, with all the heavier meanings of savior-of-the-world etc.) was supposed to anoint him with oil; and early Christians recognized John the Baptist as being that "prophet"-- but had to admit, to their embarrassment, that John had never actually anointed Jesus (John gave him water, but he gave everybody water; he didn't give him oil). A woman did rub Jesus with oil on one of the last nights before his arrest and crucifixion, however.
Would it not make sense to translate the passage this way... "Thus saith the LORD to his Messiah, to Cyrus, whose right hand I have holden, to subdue nations before him" to mean the "savior" that the Lord chose to do his bidding and not as in the one who saved the Lord?
This would be reasonable (at least, I think so). I do not know for sure if Cyrus had an oil-ritual when he was crowned king, but since this was in common between Israel and India, I would think it probably happened in Persia too. But the point is that the ritual stood for GOD anointing the king, not just the human who was rubbing the oil, and the passage in Isaiah is saying that YHWH (the Jewish God) is indeed endorsing Cyrus and using Cyrus for His work.
Also does anyone know what the Greek and Latin equivalents for "anointed" were in their respective versions of the Hebrew Bible?
The Greek is
christos, same as "Christ"; the Latin is
unctus (same root as "unguent").