Was James the son of Alphaeus the presiding Apostle at the first Council held in Jerusalem?

There are three possibilities about who James is.

1. Brother of the Lord (same mother, different father)
2. Step-brother of the Lord (different mother, different father)
3. Cousin of the Lord (son of Alphaeus)

We will look at each choice.

Two, James the step-brother

The Orthodox Church generally believes this explanation.

The step-brother of the Lord sources to the Infancy Gospel of James. In this apocryphal source, the theory is posited that James was the son of Joseph who was much older than Mary and who was married prior to their betrothal.

And the priest said to Joseph, Thou hast been chosen by lot to take into thy keeping the virgin of the Lord. But Joseph refused, saying: I have children, and I am an old man, and she is a young girl. I am afraid lest I become a laughing-stock to the sons of Israel. Infancy Gospel of James (emphasis mine)

Three, James the Less

The theory that James was the cousin of the Lord, that James was the son of Alphaeus, rather than of Joseph and Mary, sources to Jerome. The Catholic Church and some Protestants believe this.

Jerome did not approve of the idea that only Mary was the ever-virgin. He also wanted Joseph to be an ever-virgin. Jerome believed virginity was closer to Godliness, but he also has to explain who James the brother of the Lord is.

Let me [Jerome] point out then what John says,4217 “But there were standing by the cross of Jesus his mother, and his mother’s sister, Mary the wife of Clopas, and Mary Magdalene.” No one doubts that there were two apostles called by the name James, James the son of Zebedee, and James the son of Alphæus. Do you intend the comparatively unknown James the less, who is called in Scripture the son of Mary, not however of Mary the mother of our Lord, to be an apostle, or not? If he is an apostle, he must be the son of Alphæus and a believer in Jesus, “For neither did his brethren believe in him.” Jerome, Against Helvidius (emphasis mine)

So, Jerome here asserts how Catholicism and most of Protestantism understands who James the Less is; that is, he is the son of Alphaeus.

Jerome further explains how the word "brother" may be used in scripture.

How then, says Helvidius, do you make out that they were called the Lord’s brethren who were not his brethren? I will show how that is. In Holy Scripture there are four kinds of brethren—by nature, race, kindred, love. Jerome, Against Helvidius

From there, Jerome makes his primary point.

I [Jerome] claim still more, that Joseph himself on account of Mary was a virgin, so that from a virgin wedlock a virgin son was born. Jerome, Against Helvidius

One, James the Just

The alternative to the step-brother or cousin theories is that Mary and Joseph had children after the birth of Christ Jesus.

Against Helvidius was written about 380 CE. Much later than apostolic times. The Infancy Gospel likewise was written as early as 150 CE, but most like about 250 CE. These are late theories.

The main problem with the Infancy Gospel is it teaches that Mary had a midwife to help her. This contradicts scripture. It also teaches Jesus was born in a cave, rather than a manger (Luke 2:7). It also mentions the water of jealousy, which could induce an abortion. The Catholic Church thus rejected this book as spurious. This from here about the Infancy Gospel "despite being condemned by Pope Innocent I in 405 and rejected by the Gelasian Decree around 500, became a widely influential source for Mariology."

With that notion out of the way, an alternate explanation of who the brothers were became necessary. Enter Jerome's cousin theory.

The alternative is simply to agree with scripture and others of the early church.

Is not this the carpenter, the son of Mary, the brother of James, and Joses, and of Juda, and Simon? and are not his sisters here with us? And they were offended at him.

When alternatives are dismissed, we may also look for other explanations. The earliest reference, besides scripture, is captured from Eusebius. The brothers were "according to the flesh", of the house of David.

1. “Of the family of the Lord there were still living the grandchildren of Jude, who is said to have been the Lord’s brother according to the flesh.
2. Information was given that they belonged to the family of David, and they were brought to the Emperor Domitian by the Evocatus. For Domitian feared the coming of Christ as Herod also had feared it. And he asked them if they were descendants of David, and they confessed that they were. Eusebius Book III Chapter 20

James the Just would be the first bishop of Jerusalem.

1. But after Paul, in consequence of his appeal to Cæsar, had been sent to Rome by Festus, the Jews, being frustrated in their hope of entrapping him by the snares which they had laid for him, turned against James, the brother of the Lord,484 to whom the episcopal seat at Jerusalem had been entrusted by the apostles. ...
2. Josephus, at least, has not hesitated to testify this in his writings, where he says, “These things happened to the Jews to avenge James the Just, who was a brother of Jesus, that is called the Christ. For the Jews slew him, although he was a most just man.” Eusebius Book II Chapter 23

So, to directly answer the OP, James the Just, brother of the Lord Jesus, was the first presiding bishop of Jerusalem.