Christian versions of the Bible seem to have a different versing scheme than Jewish ones for Genesis chapter 32. What is the first verse in the Christian versions is the last verse of 31 in the Jewish versions. This means that the verses of chapter 32 are one off between the versions.
What is the history here? Not just of the idea of differences in general, but how or at what point this chapter got divided differently. And what old versions use which numbering system?
By starting chapter 32 one verse earlier, at (NIV):
Early the next morning Laban kissed his grandchildren and his daughters and blessed them. Then he left and returned home.
the Jewish chapter division leaves the last three verses of a parsha together at the start of chapter 32 before the open parsha break at
Jacob sent messengers ahead of him to his brother Esau in the land of Seir, the country of Edom.
which is also the start of a weekly reading. Not leaving less than three verses at the end or beginning of a parsha is a gesture of respect to the text as elucidated by a breita in Babylonian Talmud tractate Megilla page 22a1:
ואמר רבי תנחום אמר רבי יהושע בן לוי: כשם שאין מתחילין בפרשה פחות משלשה פסוקים כך אין משיירין בפרשה פחות משלשה פסוקים
This breita comes from a time before the printing press and before the chapter breaks, and refers specifically to the way the text is to be read out loud from a scroll in public readings. By the time of printing, this ingrained sensibility was extended to the way the text was printed, at least in instances where no major inconvenience was caused.2
"Rabbi Yehoshua Ben Levy also related in the name of Rabbi Tanhum: In the same way that we don't finish a reading less that three verses into a parsha, we also don't stop reading leaving less than three verses to the end of a parsha."
This answer is based on my own understanding. I do not have any historical or scholarly references for it, i.e. it's a shot in the dark.
I'm not sure why you contrast Hebrew and Jewish versification; they are one and the same. In Hebrew/Jewish Bibles, Genesis 31:54 is "ויזבח יעקב זבח בהר ויקרא לאחיו לאכל לחם ויאכלו לחם וילינו בהר". In the translation of the NRSV, this is rendered: "and Jacob offered a sacrifice on the height and called his kinsfolk to eat bread; and they ate bread and tarried all night in the hill country". (For what it's worth, the Hebrew term that NRSV is translating "on the height" in one instance and "in the hill country" in the other is one word: בהר).
In Hebrew/Jewish Bibles, that's the final verse of chapter 31. Genesis 32:1 is "וישכם לבן בבקר וינשק לבניו ולבנותיו ויברך אתהם וילך וישב לבן למקמו". In the translation of the NRSV, where this is Genesis 31:55, it is rendered: "Early in the morning Laban rose up, and kissed his grandchildren and his daughters and blessed them; then he departed and he returned home." In the NRSV, Genesis 32:1 reads "Jacob went on his way and the angels of God met him"; in Hebrew/Jewish Bibles, this verse ("ויעקב הלך לדרכו ויפגעו בו מלאכי אלהים") is Genesis 32:2.
So far as the testimony of ancient witnesses is concerned, this is a bit complicated. Printed editions of the Samaritan Pentateuch follow the masoretic versification, but older manuscripts don't feature supralinear notation that might reveal to us whether or not their traditions and the Jewish traditions departed from one another on these issues. As such, if you consult the versions of Mark Shoulson or Avraham Tsedaqa, you will see that they match modern Hebrew Bibles in their division and arrangement of the text, and not Christian Bibles.
Unfortunately, this particular passage doesn't appear to have been preserved amongst the Dead Sea Scrolls. Of the 20 fragments of Genesis and of Genesis/Exodus, none feature this particular part of the text. But even if they did, there is no accurate way of estimating from a Qumranic scroll what the versification might be, and printed editions tend to follow the Masoretic Text as well.
It stands to reason that Targums follow the Masoretic Text also, having been preserved by Jewish communities, but the exception (if we look upon it as a Targum of sorts) is the Peshitta. In the Syriac, Genesis 32:1 appears as follows: "ܘܐܦ ܝܥܩܘܒ ܐܙܠ ܠܐܘܪܚܗ ܘܦܓܥܘ ܒܗ ܡܠܐܟܘܗܝ ܕܐܠܗܐ". (In the characters normally employed for Aramaic, this is: ואף יעקב אזל לאורחה ופגעו בה מלאכוהי דאלהא). This means, "Then Jacob went on his way and the angels/messengers of God encountered him" - in other words, what the NRSV records as 32:1 and the Hebrew Bible records as 32:2.
The LXX does the same thing, by the way, although it's a somewhat longer verse. In the translation of the Zondervan Publishing House, Genesis 32:1 reads: "And Jacob departed for his journey; and having looked up, he saw the host of God encamped; and the angels of God met him". So too the Vulgate, albeit closer to the Hebrew in meaning.
In any case, none of this should be surprising. Given that the division of the text into chapters is a later development than was the composition of any one of these texts, for whatever reason Jewish and Christian versions differed on this one point, texts preserved by Jews (MT, Targums) all follow one tradition and texts preserved by Christians (LXX, Peshitta, Vulgate) all follow the other. For texts that precede the division into chapters, there is no way of knowing whether they more naturally lend themselves to one division or the other.
