That is a terrific and kind of complicated question. The best way to answer it is to go to a specific passage. In this case, let us go to Exodus 3:1-6. It is clear from the text that the person who appeared in the burning bush is both the angel of the Lord and God himself. Yet the two are distinguished, aren't they? It is helpful to know that the word angel means messenger in both Hebrew and Greek. So the person in the bush who appeared to Moses was a messenger sent by God and God himself. We know from the doctrine of the Trinity that there is one God in three persons, Father, Son and Holy Spirit. We also know from Scripture that quite often the name God is used for the Father. This distinction is also present in the New Testament, and is used by Jesus himself in his conversation with Mary Magdalene in John 20:17, "Jesus said to her, Do not cling to me, for I have not yet ascended to the Father; but go to my brothers and say to them, 'I am ascending to my Father and your Father, to my God and your God.'"
So what we have in Exodus is a messenger, who is God, appearing to Moses. It is reasonable to conclude that one of the persons of the Trinity appeared to Moses, and that it was not the Father. Neither would it be the Holy Spirit who is not really spoken of clearly in the Old Testament, though he is spoken of as the Spirit of God. The only person left is the Son.
Now, the man, Christ Jesus, did not come to be born, i.e., the Son of God did not become man, until he was born in Bethlehem. So what we have in Exodus 3:1-7 is the Son of God in a pre-incarnate appearance. You can say it is Jesus, of course, because you understand Jesus is the Son of God. But you are not calling him an angel in the sense that he is like Gabriel or Michael. Rather, he is God's messenger to Moses. You have similar appearances in other places in the Old Testament such as Joshua 5:13-15 and Daniel 3:25.
I hope this rather extended study is helpful to you.
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