I’m not sure what to say exactly, but the Wikipedia article, while rough around the edges, is not a bad summary.
The basic point is that there are multiple accounts of the Council, all found in the canonical Vinayas, and hence as historically reliable as can be at such a distance. Lacking confirmation in archeology or non-Buddhist sources, we are left with the Vinaya accounts (supplemented by later commentaries).
These accounts agree in the main, but differ in many details, such as the order of the texts, or whether the Abhidhamma was recited.
In some cases we can assess these things independently. We know from textual analysis that the Abhidhamma was later and could not have been recited at the First Council. So this detail must be discounted. Does it mean that versions of the story that mention the Abhidhamma are less reliable? Not in and of itself, no. It is normal for such narratives to include both historical details and a degree of elaboration. Different narratives might elaborate in different ways.
A detail such as the order in which the recitation was done is harder to assess. It certainly stands as a clue of the perspective of the redactors of that account. But does it tell us anything about what actually happened? It might do, but it’s hard to say with any confidence.
The accounts of the Council are careful to record the existence of a tradition that was not present and rejects the council authority. And I think it is important to recognize that in a decentralized body such as the early Sangha, a narrative of central control is a record of the wishes of those making the narrative, not a description of the facts on the ground. Clearly the Councils, in fact and idea, exert a large influence on the Buddhist tradition. equally, though, there would have from the beginning a diversity of views, some of whom would contest the central authority. In fact, what we witness with the First Council is the first step toward establishing the very idea of a centralized authority in Buddhism.
Having said which, bear in mind that the items contested, when they are actually mentioned in the Vinaya, are extremely minor, relating to the relaxing of certain Vinaya rules about food in the time of famine. Later there were disputes about more serious matters, but we should be careful to not read later situations back into early times.
Even a question such as whether the Anguttara or the Samyutta was recited first may be interesting, but it’s hardly a dramatic difference. There’s no question about the main content of the teaching.
It is unreasonable to question or dismiss the historicity of the First Council based on minor differences; and equally unreasonable to insist that one’s one particular sect just happens to include the One and Only True View. As in all things historical, balance, sympathy, reasonableness, and humility are our best guides.