Regarding the difference in names, we recognize that the Disciples generally went by two different names: one Greek, and one Aramaic (or Hebrew). Simon (bar-Jonah), for example, was also known by his Greek name "Petros" (an interesting anglicization for Peter could have been "Rock Johnson"); Levi was also known as Matthaios, anglicized as "Matthew". Although the Disciples Aramaic/Hebrew names are frequently given, they are generally referred to in the New Testament Scriptures - which are written in Greek - by their Greek names.
This explains how Matthew is named differently in the Gospels, but it doesn't answer your question as to why specifically in Matthew's Gospel he is named with his Greek name.
The consensus of the Church Fathers regarding the disparity in names in these passages is that the Luke and Mark did not wish to call attention to Matthew's sinful past as a publican. One finds that when Luke and Mark refer to Matthew elsewhere they use his Greek name (Mark 3:18, Luke 6:15), but when they are referring to the time when he was still a publican they refer to him as "Levi" (Mark 2:14, Luke 5:27,29). John Chrysostom (ca 349-407) notes, "And we have cause also to admire the self-denial of the evangelist, how he disguises not his own former life, but adds even his name, when the others had concealed him under another appellation."1 Theophylact, a later Byzantine commentator (ca 1055-1107) explains:
Marvel at how the evangelist displays his own former way of life, while the other evangelists disguise his name, calling him "Levi".2
A similar interpretation is found in the writings of the western Fathers. Jerome (ca 347-420) writes, for example:
Out of respect and deference, the other Evangelists were unwilling to call him by the common name of Matthew but said Levi. So Matthew went by a double name in accordance with what Solomon noted: An accuser is righteous at the beginning of his words.3 And in another place: Tell your sins, and you will be justified.4 Matthew also calls himself a publican to show his readers that no one must despair of salvation if he has changed for the better, for he was suddenly changed from a publican to an apostle.5
1. Homily XXX on Matthew (tr. from the Greek)
2. Explanation of the Holy Gospel According to St. Matthew (tr. from the Greek; Chrysostom Press, 2008, p.77)
3. Proverbs 18:17 LXX
4. Isaiah 43:26 LXX
4. Commentary on Matthew I.IX.9