Grenz adds to the palaeographic argument some discussion of the numbering of
the Psalms. He notes that the first numerating hand (my Nl, his Old-NumH) began
an internal numbering of the Psalms that started in Psalm 1 but for some reason
Subsequent versions of the Psalms have addressed these differences in a number of ways. Saint Jerome’s translation of the Bible into Latin (the Vulgate), for instance, follows the Septuagint numbering system. For many years, Roman Catholic translations of the Bible into English followed Jerome’s numbering, although more recent Catholic Bibles have reverted to the Hebrew designations. Protestant Bible translations have almost always followed the Hebrew numbering of the Psalms; thus, contemporary Protestant translations accord with the numbering of contemporary Catholic translations. The Orthodox Church continues to follow the Greek system, meaning that contemporary Orthodox translations of the Bible number most Psalms differently than contemporary Protestant and Catholic Bibles (this may be important to remember if you ever hold an ecumenical Bible study).
This confused me a bit, it sounded like 3 systems, but it is 2.
Charles Sullivan https://charlesasullivan.com/11545/the-structure-of-the-psalms/
► The Greek texts
These ancient Christian manuscripts which contained the Book of Psalms were examined:
Codex Vaticanus 1209 (@315AD),6 Codex Sinaiticus (@345 AD),7 and Codex Alexandrinus
(@420 AD).8
Both the Vaticanus and Sinaiticus has each Psalm numbered. The Sinaiticus goes a step further and colors the header text in red. The Codex Alexandrinus has all the Psalms numbered but the headers are not so easy to distinguish. The physical condition of Codex Alexandrinus is lower than the two mentioned above. Codex Alexandrinus lacks the aesthetic beauty of its two other counterparts.
All of them emphasize the chorus line called a dipsalma (a pause between two verses so that different parts of the choir could property harmonize in singing the Psalm). It is clear that the numbering system in use with the Psalms today comes from the Septuagint but it is unclear when the Greeks started this tradition. We know it began before the 300s because both the Codex Vaticanus and Sinaiticus contains them. However, none of the New Testament writings contain any numbered Bible references. Similarly, the Jewish Aramaic community does not cite such a tradition either throughout the Mishnah or Gemarrah - the two main pieces that constitute the Jewish Babylonian Talmud.