The idea that Christ’s (one) being consisted of two natures is known in theological circles as “hypostasis” (i.e. the “hypostatic union”). Developing a solid understanding of this idea in relation to its intellectual competitors (like the Nestorian and Monophysite heresies that you mention above) is best done, I would argue, by considering the issue within the broader context of then-ongoing debates that were concerned with articulating the nature God-as-trinity and with fitting the figure of Christ into that trinitarian equation.

The reason that so many theologians of the early Christian centuries (Origen, Gregory of Nyssa, Tertullian, Athanasius, and many others) were so concerned with describing the nature of Christ in such a very particular and philosophically precise manner is explainable primarily by their having been faced with the problem of needing to reconcile their understanding of Christ as possessing some form of divinity with a number of (at least apparent) contradictions between that understanding and their more general understanding of God. As Maurice Wiles points out in the notes of his extremely useful collection “Documents In Early Christian Thought” (Cambridge University Press, 1975), there existed a need for coherently combining “belief in the unity of God with the ascription of divinity to Christ, and also to the holy spirit.” (22)

Given that this was the task at hand, it follows that a major reason why certain Christologies came to be rejected as heretical was largely due to the fact that they came to be viewed as philosophically incapable of achieving this task for one reason or another. Without splitting hairs over the painstaking details of the various Christological positions that existed in the early Church, it is worth remembering the particular meaning of the term “heresy”- it essentially refers to an idea that is at odds with the orthodox or majority position of a given issue. All of the various Christologies that were floating around in the early centuries of Christianity were offered up (more or less) as attempts at resolving the theological problems that surrounded ideas about Christ’s relationship with God, and the position that came to be viewed as the orthodox one (hypostasis- Christ as “fully God and fully man” rather than as some lesser combination of the two or as not really God or not really man) was collectively deemed to achieve this aim more precisely than competing Christologies.

This orthodox position of the hypostatic union was established most formally during the Council of Chalcedon in 451, in no small part due to the influence of Leo, Bishop of Rome from 440-461. As Wiles again points out, Pope Leo played a significant role in “ensuring that the Council of Chalcedon did full justice to both natures (of Christ)” (44). The text of Leo’s famous Tome articulated a nuanced sense of the position that came to be adopted by the Church at large. A more thorough response to this question would require a deeper dive into philosophical and theological concepts that I think start to go beyond the scope of historical analysis (and at the very least, start to go beyond the scope of what I feel competent speaking about), but if you take a look at the (pretty readable) text of Leo’s document that I linked, that should help you to get a better sense of the intellectual, Christological themes in play at the point of the Council of Chalcedon.