It is not contradictory since the meaning of "second death" in Rev 21:8 is uncontroversial: death because of punishment after the Day of Judgment, not another earthly death after a soul has been reincarnated. "Second death" has a very different nature than the earthly "first death" that has been or will be experienced by everyone while on earth (except Elijah, Enoch, etc.).
From a 2017 blog article by a theology professor Terrance Tiessen First and second death: similarities and differences:
Christians have uniformly affirmed the resurrection of everyone at Christ’s return, and there has been no controversy over what this means for those who died in faith, regardless of the covenant under which they died. Believers, even those whose faith was necessarily pre-Christian (that is, whose response to God was assessed according to an earlier covenant than the new one), are raised with Christ, to enjoy life in the new heaven and earth that God establishes after the final judgment. But unbelievers do not participate in the redemption accomplished by Christ, and they are excluded from the gloriously renewed creation. Instead, their destiny is Gehenna, hell, the consuming lake of fire, shame and contempt. By contrast with the righteous in Christ, who are given immortality and eternal life, those who are condemned are punished with the second death (Rev 2:11, 20:6, 14, and 21:8).
Terrance also said something about how Jesus's resurrection is NOT reincarnation:
It is certain that Jesus was not personally extinguished in his humanity, in death, and then recreated (reincarnated) 3 days later. That would entail 2 incarnations and be a highly irregular Christology. On the other hand, it is important that we not treat Jesus’ situation as unique. In his experience of death and resurrection for us he went through the process that we who are “in Christ” also go through, thereby transforming the meaning of death for us and assuring us of the resurrection to glory. It was the eternal Word, who was pure spirit, who took upon himself human nature (i.e., he added to his essential divine attributes the essential human attributes). What matters for us here, is the continued personal existence of Jesus as both God and man, after his bodily death, which, in the manner Jesus experienced death as a human being, was no different from that of every other human being. As the Definition of Chalcedon put it, in being begotten of the virgin Mary, the eternal Son of God became a man “like us in all respects apart from sin,” and this pertained equally to his experience between death and resurrection.
Another article Does the Second Death imply Reincarnation provides a typical exegesis of the meaning of "second death" (emphasis mine):
When Jesus spoke to the church at Smyrna (Revelation 2:8-10) and warned of future trials they would face, he encouraged them to be “faithful unto death,” that is, possess the martyr spirit that was essential to their being victorious with the Lord (cf. Revelation 6:9-11; Revelation 7:13-15). Jesus assures them that the ones who “overcome” will not be hurt by “the second death” (Revelation 2:11 – KJV).
Note that the associated word “hurt” necessarily says something about the meaning of the phrase, “second death.” The use of this phrase does not allow the idea that there will be another chance, just in case we don’t “get it right” the first time. It is identifiable with what “hurts” men, and is relative to some kind of punishment not what is beneficial to man.
This phrase does not appear again in Revelation until Revelation 20:6 and Revelation 20:14. In this text, let us focus on “the second death” as it used with reference to punishment.
This time, it is identifiable with the eternal state of the enemies of Christ death (the enemy of man 1 Corinthians 15:26) and Hades (the unseen abiding place of the dead) are cast into the lake of fire as are the beast (the corrupt, persecuting political power) and the false prophet (false religion) note Revelation 20:10.
Then noting Revelation 21:8 we read, “But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.”
Those who believe in reincarnation really do not want to use this verse, because it clearly speaks of “the second death” as a place of irrevocable punishment the lake that burns with fire and brimstone (Revelation 20:10).