At what moment did humans become sinful?

I don't think that it is strictly correct to say that "Eastern Orthodox Christians view the matter of sin, the sin is divided by them into three categories: 1) original sin, 2) hereditary or generational sin, and 3) personal sin." In general, Eastern Orthodoxy is not nearly this taxonomic.

What is Sin?

One of the better explanations of the Orthodox understanding of sin comes from the late Archbishop Dmitri Royster's commentary on Romans, St. Paul's Epistle to the Romans: A Pastoral Commentary. Commenting on Romans 1:18 (For I am not ashamed of the Gospel of Christ: for it is the power of God unto salvation to everyone that believeth; to the Jew first, and also to the Greek), he writes:

What does "being saved" mean? From what sins do men need to be saved? Since sin in the Greek original is hamartia, literally "failure" or "missing the mark," we have to conclude that man's sin consists in fundamentally missing the very point in his existence (although for some Christians, salvation has been reduced to nothing more than escaping the punishment of hell).

This is the context for understanding the precise definition of sin given by the Evangelist John in his first Epistle (3:4):

Sin is lawlessness

The Greek here is anomia: the root being the Greek word for law (nomos) with the prefix "a-" indicating a lack of. The "law" being referred to here, though, is not the law of Moses, but rather that law that came into man's hearts through the new convenant that was prophesied by Jeremiah (31:33; also Hebrews 10:16).

What is "Original Sin"?

The understanding of "original sin" in the eastern Church is radically different form the understanding in western Christian confessions (i.e. Roman Catholicism, Protestantism). The difference largely arises from a fundamentally different understanding about the nature of grace itself and the impact of the Fall on man's ability to experience/receive that grace.

The two views are contrasted by Abbot Damascene in his notes to the 3rd English Edition of Protopresbyter Michael Pomazanski's Orthodox Dogmatic Theology. Regarding grace, Abbot Damascene writes:

According to the Catholic Encyclopedia, “Original sin is the privation of sanctifying grace” (1911 edition, vol. 11, p. 314). As noted above ... the Orthodox Holy Fathers also teach that man lost the Grace of God at the time He fell. However, the Orthodox teaching on this subject is different from the Roman Catholic teaching in two ways.

First of all, in Roman Catholic teaching grace is a created phenomenon: “It is not a substance that exists by itself, or apart from the soul; therefore it is a physical accident inhering in the soul.... Sanctifying grace may be philosophically termed a ‘permanent, supernatural quality of the soul’” (Catholic Encyclopedia, 1911 edition, vol. 6, p. 705). According to Orthodox theology, on the other hand, Grace is the Uncreated Energy of God Himself, which at the time of man’s creation was intimately connected with his soul. Man participated in the Divine life through the Divine Energy, and this participation was proper to the original nature of man.¹

This understanding of grace leads to a much different interpretation of the Fall:

Secondly, in Roman Catholic teaching original sin consists only in the privation of sanctifying grace (also called “original justice”), while the nature of man remained the same after the fall as it had been before the fall. In this view, the nature of man has not become corrupted; rather, the privation of grace in itself constitutes “a stain, a moral deformity” (Catholic Encyclopedia, vol. 11, p. 314). According to Orthodox theology, on the other hand, man’s nature was corrupted at the fall, and this corruption caused man to lose the indwelling of Grace and deprived him of participation in God.²

As Vladimir Lossky, noted 20th century Orthodox theologian, wrote:

The deprivation of Grace is not the cause, but rather the consequence of the decadence of our nature.³

Original Sin, Hereditary Sin

Addressing your individual comments/questions:

The original sin is the sin committed by Adam and Eve. In other words, it's their act of not following upon God's prohibition on partaking of one particular tree in the garden of Eden.

The hereditary sin is that damaged state that all humans have ended up in due to Adam and Eve's original sin.

The term "original sin" is generally avoided within the Orthodox Church, since it invites confusion with Roman Catholic theology. Instead, the term "ancestral sin" is almost used in English Orthodox texts.

First and foremost, ancestral sin is understood to be a disease. It is definitely not understood as a sort of juridical guilt that gets inherited. Cyril of Alexandria (378-444) provided the following explanation:

What has Adam’s guilt to do with us? Why are we held responsible for his sin when we were not even born when he committed it? Did not God say: The parents will not die for the children, nor the children for the parents, but the soul which has sinned, it shall die (Deut. 24: 16)? We have become sinners because of Adam’s disobedience in the following manner.... After he fell into sin and surrendered to corruption, impure lusts [or pleasures] invaded the nature of his flesh, and at the same time the evil law of our members was born. For our nature contracted the disease of sin because of the disobedience of one man, that is, Adam, and thus many became sinners. This was not because they sinned along with Adam, because they did not then exist, but because they had the same nature as Adam, which fell under the law of sin. Thus, just as human nature acquired the weakness of corruption in Adam because of disobedience, and evil desires [or passions] invaded it, so the same nature was later set free by Christ, Who was obedient to God the Father and did not commit sin.

Abbot Damascene also elaborates on the Orthodox understanding of ancestral sin:

The Eastern Orthodox Holy Fathers often affirm that all of Adam’s descendants inherit his sin, in accordance with the words of St. Paul: By one man’s disobedience, many were made sinners (Rom. 5: 19). However, in saying this they do not mean that the guilt of Adam’s sin was imputed to his descendants; rather, it was the consequences of that sin that were transmitted. These consequences, as we have seen, include suffering, death, and physical corruption; a corruption of human nature; and a consequent loss of the indwelling Grace of God.⁵

The Precise Moment ...

Although we use the term "sin" interchangeably, there is a difference between the state, or better, disease, of sin, and a sinful act. We refer to the former as "sin", and to the latter in terms of "sins".⁶

So perhaps in some context it makes sense to talk about an Original Sin or the Original Sin, that is not Orthodox thinking, I think. But I think I understand the motivation for your question:

At what moment did humans become sinful? Precisely, at what moment did humans acquire sinful passions?

The way that the question is worded makes it seems evil seem more substantive than it is. Passion, in the understanding of the Orthodox Fathers, is captivity to sin. In the same way we would not ask about how someone acquired incarceration, I think we would not ask how one "acquired" passions. Passion is complete surrender to sin.

Archbishop Averky Tauchev explained the stages leading to the passions in his book, The Struggle for Virtue: Asceticism in a Modern Secular Society:

Man’s falling into sin takes place in accordance with a certain consistent pattern. The first stage of sin is the stage of “suggestion,” when sinful thoughts and suggestions enter unintentionally, by chance, and contrary to one’s will into one’s soul, either through the senses, the emotions or through the imagination. This is sinless and is only a possible prelude to sin. “Acceptance” is the reception of a “suggestion,” paying heed to it, which is not always without sin. “Consent” is when the soul delights in the thought or image that has been presented; at this point, there is the danger of actually falling into sin by deed. The next stage is “captivity,” when the soul is so strongly drawn to sin that the peaceful state of the soul is disturbed. Finally comes “passion,” the long-term and habitual delight in sinful thoughts and feelings, and the committing of sin in actual deed. This is already complete slavery to sin, and the one who does not repent and drive out his passion will be subject to eternal torment. However, someone possessed by one or another passion begins to experience a foretaste of eternal torment already in this life and will not find inner peace. At this point a very intense and persistent battle and the special Grace of God is necessary in order to renounce a sin that has become second nature.⁷

In Genesis, we see these stages played out:

• First we have the suggestion of the devil (Genesis 3:1)
• Next, we have an acceptance of the devil's suggestion by Eve (Genesis 3:2). The right thing for Eve to have done would have been to not even converse with the serpent.
• Third, we have further discourse and Eve's consent to the suggestion (And when the woman saw that the tree was good for food and pleasing to the eye ...; Genesis 3:6) and becoming captive to the thought

The passion is the actual committing of the deed of not only eating of the fruit herself, but also giving some to her husband (Genesis 3:6). So if one wanted to identify some sort of watershed event, this might be it.

Even this, however, is not necessarily true. The Church Fathers understood that the events that immediately transpired after the transgression were God's giving Adam and Eve the opportunity to repent. First they heard the voice of the Lord God walking in the garden ... (Genesis 3:8). "What do you say?" wrote John Chrysostom (c 349-407), "Are you going to ascribe feet to Him?"

Pay heed to the Lord's love of mankind and His extreme lack of ill will. He could, without even vouchsafing a reply to the one who had performed such a sin, have immediately subjected him to the punishment ... but He is long-suffering, delays, asks and listens to the answer, and again asks [Genesis 3:10-13], as if evoking the guilty one to justify himself in order that the matter might have been revealed He might show him His love of mankind even after such a transgression.⁸

Thus, Adam's and Eve's failure - in the understanding of the Church Fathers - was perhaps not primarily their transgression, but rather their subsequent failure to repent when offered the occasion to do so. As Ephraim the Syrian (306-373) commented:

If our first ancestors had desired to repent even after the transgression of the commandment, then, even though they would not have restored to themselves what they had before the transgression of the commandment, at least they would have been delivered from the curses that were uttered to the earth and to themselves.

^{1 p. 166n
2 p. 166-67n
3 The Mystical Theology of the Eastern Church, p.132.
4 Commentary on Romans 5:18
5 Orthodox Dogmatic Theology, p.163n
6 Most translations of the Masoretic Hebrew of Psalm 51 (part of Orthodox daily prayers) read something like I was conceived in iniquity and in sin did my mother bear me, but the Greek Septuagint version of the Psalm (about 1,000 years older than the Masoretic Text) actually reads I was conceived in iniquities and in sins did my mother bear me - an interesting distinction.
7 p.133-134
8 Homilies on Genesis}