I don't think a website like that is the best resource for church history - all the dates are mistaken for instance! Justinian did condemn Origenism in 543, but it was done with the pope's consent - Origenism had no friends in the west, and only a small group in Palestine was pro-Origen. Vigilius was imprisoned and forced to sign up to an ecumenical council, but that was in 553 over a very different and a much more urgent issue, the condemnation of various Chalcedonian treatises that were seen by some as too Nestorian. To be fair, the Council of 553 condemned Origenism as well, but the academic consensus is that because the pope was disagreeing with the emperor, the council ended up doing some preliminary business to fill up the time - Origenism was an easy and convenient target. Moreover, Origen had been controversial since the fourth century, so Justinian's condemnations of Origen in 543 and 553 were mostly likely attempts to shore up his orthodox credentials rather than anything innovative.

If you want to read the original texts from 543 and 553 for yourself, Richard Price's translation and commentary available in The Acts of the Council of Constantinople of 553 (2009) is excellent. It is pretty difficult to read, but unfortunately I can't think of any introductory texts to such an esoteric topic. As for the broader issue of reincarnation in Christianity, I'm not qualified enough to comment, so hopefully someone else can answer that part of your question!

The idea that early Christians believed or taught re-incarnation and that this was later ‘stamped out’ is common, in my experience, primarily on the internet.

The not-very-accurate webpage you linked to discusses this with reference to Origen. Origen is the main source brought forward for the reincarnation view, so let’s talk about Origen. Origen (ca. 184-253) is the most influential theologian in the Greek-speaking milieu from Paul onwards, so much so that everyone read Origen. They didn’t necessarily agree with him, and sometimes different parts of Origen headed later theologians off in different traditions.

We should also just stop and point out that there are different ‘versions’ of reincarnation. Do souls reincarnate only as human beings? Or is reincarnation as a different life form (as per Pythagoras, as well as eastern religions), possible? Or, is it that there are ‘ages’ of the universe and that souls get an incarnation ‘per age’?

In Against Celsus 1.32, Origen discusses the idea of a body fitting for a soul according to the soul’s past actions, as part of a counter-argument against Celsus, but specifically says, “I speak now according to the opinion of Pythagoras, and Plato, and Empedocles, whom Celsus frequently names”. ie. his counter-argument employs the philosophical bases for the sake of argument that Celsus has been utilising.

A second passage often cited comes in AC 7.32, where Origen specifically tells us that his teaching on resurrection is not derived from a doctrine of metempsychosis (transmigration of the soul).

It is quite difficult to assert with clear textual evidence that Origen taught reincarnation, and fanciful to assert it with reference to other authors. A few other passages are sometimes cited, but they are even weaker and I will omit them for the moment. So why the claims about Justinian and the condemnation of Origen? In the late 4th century a ‘controversy’ broke out over Origen’s theology and legacy. This, like most 4th and 5th century controversies, interwove theological and ecclesial-political issues. It was in part caused by a polarisation with a faction that tended towards anthropomorphism and hyper-literalism. Anyway, the main ‘concerns’ about ‘Origenism’ included his highly allegorical exegetical approach, and the philosophical system that appears in On First Principles which includes the idea of the eternality of creation, and pre-existence of souls, as well as appears to suggest universalism (all souls will be restored to God in the end). While the first controversy didn’t result in widespread condemnation, in the 6th century Emperor Justinian did act against ‘Origenism’, but reincarnation was not the focus of this action. The second council of Constantinople in 553 was primarily concerned with the “Three Chapters” controversy and was focused on the reinforcement of Chalcedonian Christology over against tendencies towards Nestorianism. Condemnation of Origen occurred largely by the push of the Emperor, but Pope Vigilius, among others, writes of the council without any direct reference to the Origen issue. This is not proof that the Roman Pope was opposed to the condemnation, but matches with the hypothesis that the condemnation of Origen happened ‘outside’ the Council proper, possible beforehand. However, even the anathemas against Origen attributed to the Council do not describe reincarnation. Anathema one:

If anyone asserts the fabulous pre-existence of souls, and shall assert the monstrous restoration which follows from it: let him be anathema.

More likely refers to the doctrines of pre-existence and universal restoration, not to reincarnation.