The writer cites the idea that there is a Truth beyond words; and then in the very next paragraph, he betrays his lack of understanding this principle by stating, "This notion of the transmigration of the soul definitely does not exist in Buddhism." If he had really understood the principle, he would not be talking about "what exists in Buddhism," as though the religion of Buddhism contains the Truth. There is no such thing as "Buddhism." Sri Ramakrishna once was taken to a very impressive building; he said that the Divine Mother revealed to him that it was just one brick laid on another. A religion, in and of itself, is just one convention, one idea, laid on another. It's an attempt to perpetuate the memory and teachings of the Founder, who taught from the direct experience of Truth. It has its importance and its value; but no religion is as homogeneous as its followers would have us believe. This should be clear enough by the factions that arise after the Founder's departure, each claiming to be the only true representation.
So there is, really-speaking, no such thing as "what Christianity says" about reincarnation, nor "what Buddhism says" about reincarnation. The closest you could get to it is, "What did Jesus say about reincarnation?" or "What did Buddha say about reincarnation?"
Buddha did not appear to teach about God or an immortal soul, or anything that could be continuous between incarnations. Jesus did not appear to teach reincarnation (though he did refer to it in passing). Modern followers jump to the conclusion that Buddha did not believe in God or an immortal soul; and that Jesus did not believe in reincarnation. Then they take a further jump to say that "There is no reincarnation in Christianity" and "There is no God or immortal soul" in Buddhism. But where is Truth in all this? If there is only one Truth, and we accept these statements, then one of these world Teachers was flat wrong, and the other was right. We are conceptually locked in at that point. The assumptions* we have made have led us to an impasse. And wars have been fought over these impasses.
There is only one Truth. Jesus and Buddha both experienced it and taught from it. If there is a discrepancy in their teachings, then the problem lies in the incompleteness of the teachings, and the inability of human beings to intellectually grasp the Truth. The problem does not lie in the spiritual authority of either of these Teachers.
There is only one way out of this dilemma that I know of, and that is to find a contemporary teacher of the same calibre, and to use his teachings as a guide in reconciling the teachings of the teachers of the past. This requires preparation. If you are prepared, you will find it on your own. If you are not prepared, then no amount of my telling you will help. You will gravitate toward some other teacher or teaching.
Swami Vivekananda pointed out that Buddha was a reformer. I believe all the Founders of the major religions were reformers, in the sense that their teaching was an antidote for the poison of ignorance that existed in the culture into which they incarnated.** If, for example, the culture at the time of Mohammed was given to idolatry, then his teaching would be geared as an antidote for that particular spiritual poison. Buddha's teaching was also an antidote for the spiritual poison of that era. I don't know the history of that era, but you can work it backwards and see that what Buddha knew was required, was to focus on the way to Liberation, rather than to add to the intellectual baggage of concepts about God.
Jesus did not see fit to emphasize reincarnation, at least not to the masses. A clear reference to it, in parable form, survives in passage 109 of the Gospel of Thomas. Buddha taught reincarnation explicitly, but omitted the idea of an immortal soul that is continuous from incarnation to incarnation. These Teachers, each experiencing the same Truth, taught what the people at that time needed. Note that there is also the matter of editing, inadvertent corruption of texts, and deliberate distortion of texts.
Even when you find a contemporary teacher of the same calibre, and you are correct in that assessment, you will not have the entire Truth (i.e., intellectually). The zen parable of the finger pointing to the moon has been with us a long time, and people can cite it, but they don't live by it. The Hindu parable of the blind men and the elephant has been with us a long time, and people know it but their writing betrays their inability to put it into practice in their thinking. One wonders what the point of trying to communicate this is...and yet if there was no point, the Founders of the great world religions would not have bothered to teach anything.
You will find the teachings of genuine spiritual masters in the Links section of this website under the corresponding heading.
*Another such assumption is made by academicians when they suggest that this or that spiritual teaching migrated from one culture to another. The idea that there is one Truth which the Founder of each religion experienced and taught portions of isn't palatable to them, because they assume there is no such thing. They believe (and this is a philosophical stance) that truth consists of ideas that arise in a cultural context only, and therefore religious ideas had to have been transferred across cultures. Unfortunately, they may not reveal the philosophical basis for this concept when teaching it.
**I feel a "red flag" as I write this--as though it is subject to misinterpretation. It would not be correct to say that the teachings of the Founders of the great world religions were only an antidote to the spiritual poison or ignorance of the time. But it would be correct to say that this element accounts for the emphasis of the teachings--why the Founder of this or that religion emphasized certain things and downplayed other things. What I'm suggesting is that Jesus knew full-well about reincarnation, for example; and Buddha knew full-well about God and the soul. It wouldn't have been possible to teach everything--because Truth cannot be grasped intellectually, for one thing, and also because they taught from the perspective that the people of that era and culture needed. Still, Truth was embedded in the teachings, and the seeker could follow the "scent" of that Truth from the teachings to the Source in each case.