John the Baptist is not considered the reincarnation of Elijah. Luke 1:17 clearly states this when it says: “[John the Baptist] will go before [Jesus] in the spirit and power of Elijah.” John the Baptist is believed to have the role of Elijah, he is not literally the person Elijah.
Jewish people in the 1st century believed that Elijah would return to earth, not be reincarnated. Elijah had not died but was taken up to heaven (2 King 2:1-11) and thus the Jewish people believed he would physically return to earth from heaven before the coming of the messiah. That is why John the Baptist denied he was the literal physical return of Elijah from heaven but Jesus affirmed he was a “typology” (as you put it) of Elijah.
