Given that it’s New Year’s Day, and most of my books are in boxes in storage, forgive me if I am a little underprepared. The Council of Chalcedon did not decide if the doctrine of the Trinity was to be incorporated into Canon Theology. I’m not sure where you have got this idea from. The Trinity was accepted by all ‘sides’ at this council.

The council consisted of around 450 bishops. There would have been other clergy in attendance, but only the bishops were the real deal-makers at these kinds of councils. Almost all of the bishops were from the Eastern half of the Empire, apart from 2 North African bishops. The Western half of the Empire was not represented by bishops, but some delegates from the Church of Rome were in attendance on behalf of the bishop of Rome.

The council of Chalcedon really met to resolve matters that had been debated and manoeuvred through the period 381-451. The two major sides are typically polarised around a view associated with Nestorius, bishop of Constantinople from 428. Although we now think that Nestorius himself was not as ‘Nestorian’ as he has been made out to be, ‘Nestorianism’ is the label given to a view that sharply divides the divine nature and human nature in the person of Christ, to the extent that there are two persons. The opposite pole is usually associated with Cyril of Alexandria, and emphasises the union of the two natures to the degree that it seems as if there is but one, new, nature in Christ. This kind of view later finds expression in the Monophysite and similar theological movements within the Eastern churches.

The authority of this council was really a watershed in the history of councils. All the bishops present accepted that the Council of Nicaea in 325 was an ecumenical council that held binding for the whole church, but the idea of the authority of a church council was still being developed, to some extent. At Chalcedon they recognised Nicaea, Constantinople (381), and Ephesus (431) as authoritative councils, and they clearly had a sense that this council was to join that number. The council dealt with a number of matters, but in regards to the two-natures controversy, it basically accepted a position on a trajectory of Nicaea (325), Constantinople (381), Ephesus (431) and Cyril’s position there, and Pope Leo’s Tome, which expressed a two-nature one-person position. This council then produced what is known as the Chalcedonian Definition.

follow-up questions?

Who would I likely be (Who attended this meeting, how large was it?)

Contemporary letters suggest that up to 500 bishops attended the Council, but based on the actual record of the Council it seems like not every bishop attended every meeting and that some metropolitan bishops voted for bishops under their authority. In Richard Price and Michael Gaddis's recent translation and commentary they suggested a figure of around 370 for attendees with voting rights. There were of course many lower-ranked clergymen also present who either represented absent bishops or were just followers of the people involved. Lastly, there were also several important secular officials who chaired the Council, which necessitated the presence of their followers and guards, a mark of imperial influence made necessary given how messy the Second Council of Ephesus of 449 was. Finally, both Emperor Marcian and Empress Pulcheria briefly visited the Council, so there would have been even more people in and around the meeting during the Council's sixth session.

As /u/talondearg noted, western representation was practically non-existent apart from the delegation from Rome, which included three bishops from Italy. Only one of them knew Greek, so that means we have to add a whole bunch of translators to our list of attendees too. Four other western bishops were present, one from Pannonia and three (or two) from North Africa, but all of them can be considered to be associated with the eastern Church already and not really representing western interests (the African bishops were probably refugees from the Vandal invasion of Africa for instance). There was also seemingly a bishop from Ethiopia included as part of the Egyptian contingent.

How diverse were the opinions of those attending the Council re: incorporating the Trinity?

Everyone agreed on the Trinity at the Council. Instead, the debate was over Christology and the political fallout from the two Councils of Ephesus. To put it briefly, Nestorius and his Antiochene allies lost at the First Council of Ephesus thanks the machinations of Cyril of Alexandria and his followers. The former tended to use statements like 'two natures' (human and divine) in their theological writings, whilst the latter preferred formulations of 'one hypostasis and 'one incarnate nature' - basically some form of union between the human and divine natures of Christ. Cyril's own Christology however is also complicated, as he at times was relatively conciliatory towards to the Antiochene position, on other occasions though he was much more dogmatic, so it allowed people of all views to see the dead Cyril as their champion at Chalcedon. Nestorius himself was condemned at Ephesus I for allegedly believing in there being 'two persons' in Christ, but the Antiochene network survived.

Issues flared up again in the 440s when the Alexandrian party renewed their attack on the Antiochenes, this time led by Cyril's successor Dioscorus. To summarise a very complicated event, Ephesus II ended rather violently as Dioscorus' partisans dominated the proceedings and condemned in harsher terms anything associated with Nestorianism, in the process marginalising the sees of Antioch and Rome. By 451 there was a new emperor on the scene and Dioscorus' influence had weakened, leading some bishops to call for a new council to deal with the mess from Ephesus II. As such, the most prominent factions present at Chalcedon were Dioscorus' allies and those who thought that he had went too far. For various reasons, Dioscorus was condemned at Chalcedon and the Council sought to steer a middle course between two extremes: Eutychianism, an extreme Cyrillian view that denied Christ's humanity altogether and saw him as only of a divine nature, and Nestorianism, the view that Christ existed as two distinct persons. This resulted in the dyophysite position - that Christ had two natures in one person.

Was there a competing school of thought regarding Jesus' nature which held quite a bit of weight among Council attendees?

Yes, particularly those from Egypt and Palestine who remained loyal to their definition of the Cyrillian Christology, the belief that Christ was both divine and human in one nature (don't ask me to explain why this is different, I'm more of a church politics person!). The Chalcedonian definition on the other hand was acceptable to moderates and partisans of the papacy (as the creed was partially based on a treatise written by Pope Leo the Great). For the influence of those who lost at Chalcedon, we only have to look at Roman history for the next two centuries - non-Chalcedonians, or miaphysites, continuously resisted imperial attempts to impose Chalcedonian candidates on important sees, to the extent that eventually a parallel ecclesiastical hierarchy was established in the sixth century in Egypt and Syria. Some emperors were even sympathetic to the miaphysites, such as Zeno, Anastasius and occasionally Justinian, a trend that reached its apogee during the reign of Heraclius, who sought to reunite the various Churches within his empire even amidst the Arab Conquests, demonstrating the extent of the miaphysites' influence. Egypt, Syria and Armenia in fact would eventually become mostly miaphysite in character, which is true even today for the Coptic, Syriac and Armenian Churches.