Some clarity might come with Young's Literal Translation. First, the context of the verse in question.
A few verses earlier Daniel had been told by the heavenly messenger, Gabriel, that "From the going forth of the word to restore and to rebuild Jerusalem till Messiah the leader [is] seven weeks, and sixty and two weeks..." (Daniel 9:25). Notice how this prophecy speaks of the future Messiah as being the leader? We here have the Christ in the text, though not identified as Jesus Christ. Bear that in mind.
In Cyrus's first year as king, he issued the decree to restore and rebuild Jerusalem - 2 Chronicles 36:22 & Ezra 1:1-4. Cyrus's first year began on 24 March, 538 B.C.E. (This is according to a Nisan to Nisan Jewish calendar.) The significance of such details is that it can be worked out (with hindsight) that 483 years passed from the issuing of that decree until Jesus rode into Jerusalem, being acclaimed King.
A few years later Daniel is given a vision about a future great war. By the time of the vision, it was the third year of Cyrus after his conquest of Babylonia, and now we are in Daniel chapter 10. A glorious messenger from heaven comes to Daniel to give him understanding about the 70 years of exile now being up. Notice the way others are described by this one:
"And the head of the kingdom of Persia is standing over-against me
twenty and one days, and lo, Michael, first of the chief heads, hath
come in to help me, and I have remained there near the kings of
Persia; and I have come to cause thee to understand that which doth
happen to thy people in the latter end of the days, for yet the vision
[is] after days." Daniel 10:13-14
Daniel is able to reply to this heavenly one, speaking of him as "my lord" (vss. 17 & 19). This one adds, "Now I turn back to fight with the head of Persia; yea, I am going forth, and lo, the head of Javan hath come; but I declare to thee that which is noted down in the Writing of Truth, and there is not one strengthening himself with me, concerning these, except Michael your head." (Daniel 10:21, the verse you ask about. All italics mine.)
From this it seems clear that there are several 'heads' over nations, there are several 'chief heads,' with Michael being the chief head of those other chief heads. Michael is 'head of heads' and that is because he is head over the nation of Israel, Daniel's nation, God's chosen nation. That is why Michael is both named and declared to be Daniel's head.
However, the Messiah, the King, is the leader. He is over all the chief heads, and all other leaders. That is perfectly shown in the New Testament letter to Hebrew Christians, chapters 1 and 2. None of the angels are equal to him, for he is over all the angels, who all have to bow down to him (Hebrews 1:6). He has a throne, and a sceptre of righteousness over which to rule all the nations, and he has power over the devil, to destroy that one. No created angel (heavenly messenger) has that power! Only the Son of God, Christ the Messiah, has that power (see also Revelation 19:11-16, noting that in verse 10 an angelic messenger forbids John to prostrate himself before him; cf. Heb. 1:6).
So, Michael is certainly the prince (or, 'head') over the nation of Israel, and, therefore, Daniel's prince (or, 'head'). However, the Messiah is distinguished, even in Daniel's prophecy, hundreds of years before he made his earthly appearance, as being the leader - leader over all the heavenly messengers (angels) and 'heads' of nations. Of course, Daniel would be bone-dead before the promised Messiah was proclaimed King by his people, the people of Israel, and he would save those proclaiming him such from their sins. No created angel can save any human from their sins!