You are indeed misunderstanding this verse, not technically, because its wording seems to be quite clear, but you're interpreting it wrong, mainly because you're leaving out its innertextual context.

And as they were coming down the mountain, Jesus commanded them, "Tell no one the vision, until the Son of Man is raised from the dead." And the disciples asked him, "Then why do the scribes say that first Elijah must come?" He answered, "Elijah does come, and he will restore all things. But I tell you that Elijah has already come, and they did not recognize him, but did to him whatever they pleased. So also the Son of Man will certainly suffer at their hands." Then the disciples understood that he was speaking to them of John the Baptist. (Matthew 17:9-13)

John the Baptist here is the person that is identified with Elijah, that is quite important to bear in mind. The whole passage refers to an older belief that is already laid-out in the Old Testament. Elijah was one of the most important prophets, probably the most important prophet of the Northern Kingdom, as he vigorously fought for the exclusive worship of JHWH alone (1 Kings 17:1-7; 18). JHWH according to Elijah is the sole lord of history (1 Kings 19:15ff.) and the one who strifes for justice and damns injustice (1 Kings 21). These are key elements of early Jewish beliefs and therefore mark the importance of Elijah as a prophet of JHWH himself. According to the bible though Elijah never died, but was instead "taken" to heaven - that is very important as this bears the answer to your question.

And as they (Elijah + Elisha) still went on and talked, behold, chariots of fire and horses of fire separated the two of them. And Elijah went up by a whirlwind into heaven. (2 Kings 2:11)

This constitutes the prominent belief in early Jewish faith that Elijah might return one day (together with his last announcement of the judgement (2 Chronicles 21:12-15)), as he never died but instead was taken to God himself. The last part of the Nevi'im ("Prophets"), the book of Malachi adopts this as a prophecy of the return of Elijah upon the day of lord.

"Behold, I send my messenger, and he will prepare the way before me. And the Lord whom you seek will suddenly come to his temple; and the messenger of the covenant in whom you delight, behold, he is coming, says the Lord of hosts." (Malachi 3:1)

"Behold, I will send you Elijah the prophet before the great and awesome day of the Lord comes. And he will turn the hearts of fathers to their children and the hearts of children to their fathers, lest I come and strike the land with a decree of utter destruction." (Malachi 4:5-6)

The return of Elijah is therefore not a reincarnation, but more of a return from heaven, as Elijah was never killed, but instead taken to heaven (even though we refer to this returned Elijah as Elijah redivivus). "Heaven" in this imagery is not comparable to the popular Christian belief of "Heaven" as the dwelling place for all dead that were righteous, in early Jewish belief there is no such thing nor is there hell. "Heaven" is of course the place where God himself sits, but the dead are entering the Sche'ol, a place much more comparable to the greek Hades. To be "taken to heaven" is thus not necessarily an indicator for death and hasn't been interpreted as such within the books of both the Old and the New Testament.

Malachi is the last book in the Nevi'im and therefore marks some kind of ending, which highlights the importance of the prophecy in Malachi 4:5-6 (the last verses of Malachi). The gospels refer to this part quite a lot, as they believe John the Baptist to be Elijah redivivus, the last prophet, the messenger, who comes to earth once again to announce the coming of the lord in Jesus himself. We do not know if the historical John did indeed identify himself as Elijah redivivus or if this is a Christian interpolation of his teachings.

So, to conclude, no, this verse is certainly not an indicator for an ancient belief in reincarnation within the early Jewish faith. "Reincarnation" is not once mentioned in the Tora and did not play any role within early Jewish beliefs. It is of course discussed in the Talmud and part of the Kabbalah, but I'm not qualified enough to talk about those. "Resurrection", of course, is another matter, but that wasn't your question.