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The Life of the Blessed Virgin Mary:

In the afternoon, when her confessor again asked her for the names of the three holy kings, she answered, Mensor, the brown-faced one, after Christ's death received the name of Leander on his baptism by St. Thomas. Theokeno, the old, yellow-faced one, who was ill when Jesus visited Mensor's camp in Arabia, was baptized Leo by St. Thomas. The brown-skinned one, who was already dead when Jesus made His visit, was called Seir or Sair.' Her confessor asked her: How then was he baptized?' She answered smiling and without hesitation: He was already dead and had received the baptism of desire.' Her confessor then said: I have never heard these names in my life: how then did they get the names of Caspar, Melchior, and Balthasar?' She replied: They were called this because it goes with their character, for these names mean: (1) He goes with love; (2) He wanders about, he approaches gently and with ingratiating manners: (3) He makes rapid decisions, he quickly directs his will to the will of God.' She said this with great friendliness, and expressed the meaning of the names by making pantomimic gestures with her hand on the bed-coverlet. It must remain for the language experts to decide how far these words can be made to bear these meanings.

https://biblehub.com/library/emmerich/the_life_of_the_blessed_virgin_mary/xiii_the_journey_of_the.htm

It isn't clear whether it was directly revealed to her that Sair received a baptism of desire or whether she made the inference herself.

I would assume she wouldn't be speaking with such confidence if it wasn't revealed for she was usually shy and timid.

https://biblehub.com/library/emmerich/the_life_of_the_blessed_virgin_mary/xiii_the_journey_of_the.htm

It isn't clear whether it was directly revealed to her that Sair received a baptism of desire or whether she made the inference herself.

I would assume she wouldn't be speaking with such confidence if it wasn't revealed for she was usually shy and timid.

Meh, multiple questions.  If the one was dead before Our Lord visited (where?  presumably Limbo of the Fathers), Baptism had not been instituted yet.  What's the actual translation of "Baptism of Desire" here, as that term is a very recent one?  Emmerich's works were actually produced by Brentano, and there are many questions about how much were embellished by him.  There's also a question of the degree to which all this was "revealed" vs. the "visionary" injection her own perception of things into it.  There's a reason the Church didn't canonize her ... most likely because the Church doesn't want to have the canonization taken as an endorsement of everything in these books.  There are some direct contradictions between Emmerich and Agreda, and even a few others.  trad123 pointed one out on this very question where Mary of Agreda says that all three were baptized.  I personally suspect that neither is correct, since if I had to guess, at least 2 of the 3 were already deceased before Our Lord rose from the dead.  33 years had passed since their visit to Bethlehem, and my guess is that they were on the older side when they visited (as I doubt that you'd have a bunch of 20-somethings who would be classified as Magi).

There's also a curiosity in Emmerich about Our Lord visiting one of these in Arabia ... and there's no Tradition of Our Lord ever having gone there, whether before His Death or after His Resurrection.

Honestly... I find the book weird so far. It has already claimed that Christ was born on the 25th of November and the Annunciation on the 25th of February, both a month earlier than the Church celebrated. Furthermore, there's a weird statement that Christ was born 8 years prior to what we count from and that the first four years of Christ's life were completely forgotten. What does that even mean?

What puzzles me is that her life was full of miracles and she had the stigmata and all, her visions didn't contradict anything in Scripture even though she never read it and yet I don't get it how the Church could get all of these things so wrong. I realize why the Church both didn't condemn and didn't canonize her but I somehow expect the books would be put on the Index if they were harmful but they must've gotten an imprimatur instead.

Yes, I have little doubt about her personal holiness, but she was bed-ridden and could not write.  These books were produced by Clemens Brentano and not Katherine herself.  So, at the end of the day, who knows what percentage of it accurately reflects things the Katherine Emmerich saw and how much is Brentano's imagination and/or embellishment?  Was he interrogating her and possibly asking leading questions?  We'll probably never know in this life.  And, then, in between, one could be exceedingly holy and still be simply "mistaken" about having "seen" something that perhaps was the product of her own imagination.  Women especially are prone to more active or vivid imaginations.  St. Theresa of Avila, whenever some nun was reported to her as having visions, replied that the "visionary" should be given more meat in her diet.  So Emmerich's personal holiness did not make her immune from such mistakes, nor would such mistakes be incompatible with her personal sanctity.  And yet the Church has opted not to canonize her, and I would bet a lot that this has been primarily because the Church did not want to be seen as endorsing her books as if they were all authentic revelation, and thereby subject the Church to ridicule and mockery from her detractors as a result if some problems were later discovered with them.

Mary of Agreda's works were initially condemned by (the majority decision of a) large panel of theologians.