Reincarnation – what a tricky topic to discuss!
(1) The verse you cite clearly indicates that there is – at the very least – a similarity if not identity between the spirit and power of Elijah and that of John the Baptist.
(2) The word "spirit’ refers to the breath, the living force, what the Chinese call “chi”, what the Hindus call “prana”. So, hypothetically speaking, a portion of Elijah’s “prana” might have taken rebirth within John.
(3) The Hindus divide the human person into five different interconnected parts: the physical body, the prana/life-force, the receptive mind, the discriminative mind, and the soul.
(4) According to some Muslims who believe that the ideas of reincarnation and non-reincarnation are both true, the explanation of how both could be true goes like this: when someone dies, the physical body, prana, and mind separate from the soul. The soul continues on its journey, while portions of the prana and mind might become re-associated with a new person.
(5) Thus, one could speak of John as having portions of the the prana associated with Elijah’s soul – in some sense, you could refer to Elijah as having returned. But in another sense, one could argue that Elijah did not return, because Elijah’s soul went off to wherever it went.
(6) The Buddhists have another way of reconciling reincarnation and non-reincarnation: in each second, your body and mind are each reborn thousands, if not millions, of times. Thus, after each second, you are the same person, and yet not the same person. Thus, reincarnation (the reincarnation of exactly the same person) isn’t completely true, and non-reincarnation (the appearance of a totally, absolutely, different person) isn’t completely true either.
(7) An example often used by Buddhists is a candle flame going from one candle to another. Once you take one candle flame, and ignite the wick of a new candle, is the new flame the same or different from the old flame? Well, you could say that in some ways its the same; in other ways its different – that is, in some ways reincarnation has occurred; in other ways non-reincarnation has occurred.
(8) I haven’t even mentioned Jewish traditions of reincarnation, or "gilgul’, so I’ll leave that alone.