Neither. James the Just, bishop of Jerusalem.
For that matter, which Mary? It is unlikely that that particular Mary, the mother of James the Less and Joseph/Joses, was Mary of Nazareth, since that particular Mary was called "Mary of Clopas," (i.e. wife of Clopas) in John 19:25.
None of those. Mary, the wife of Joseph.
And Jesus is always called "THE son of Mary," but never "A" son of Mary,
"The" is a construct that is not in the original Greek.
So I think your claim about Jesus' mother being the mother of James is.... problematic.
Nope. James was universally accepted as the 1/2 brother (by blood) of Yeshua.
James the Just was and is identified with James, son of Alphaeus and James the Less, by the historic Church (Greek and Latin, East and West) and there is no evidence in the New Testament that positively establishes otherwise.
This is not to gainsay that scholars disagreed about whether this James was an older stepbrother (son of Joseph by an earlier marriage), or a cousin (son of a sister of Mary, also called Mary -- that happened, check out who was at the foot of the Cross, three Mary's).
This is still being hashed out, even in the beginning of the 21st century via a lawsuit concerning an ossuary bearing the inscription "Ya'aqov bar Yosef achui d'Yeshua" ("James son of Joseph brother of Jesus").
In that case, the Israeli judge ruled that its authenticity or inauthenticity was not proven in any way, nor that the words 'the brother of Jesus' necessarily refer to the 'Jesus' who appears in Christian writings.
That added another layer to the whole controversy, since we now had to account for multiple Jameses, multiple Marys, multiple Josephs, AND multiple people named Jesus. The Biblical Archaeological Review even said it was statistically likely that there were more than one contemporaneous men named Jesus who would have been simultaneously a son of Joseph and a brother of James.
There goes your "universal acceptance" that there was a James who was the Lord Jesus Christ's biological brother from Jesus' birthmother Mary.
This is the Mary known from antiquity as Aeiparthenos (Greek: ἀειπαρθὲ) or Semper Virgine, in both languages, literally 'Ever-virgin'.