The relevant bit of Daniel 10:3 reads as follows:
... וְהִנֵּה מִיכָאֵל אַחַד הַשָּׂרִים הָרִאשֹׁנִים בָּא לְעָזְרֵנִי ...
... but, behold, Michael, one of the chief princes, came to help me ...
OP has two intertwining interests: (1) the primary question is about the meaning here of "one" in Hebrew; but secondarily (2) how does this relate to the status of Michael among the messengers of God ("angels").
(1) The meaning of "one"
Hebrew אֶחָד (ʾeḥād) has the following range of meaning:1
- the numeral one (cardinal)
- "another" when repeated after itself (e.g. Job 41:16 = 41:8 in Hebrew)
- as an indefinite article (otherwise lacking in Hebrew; e.g. 1 Sam 1:1, "There was a man...")
- as the ordinal, "first" (especially in date formulae, e,g, Gen 8:5, "first day of the tenth month")
- as a distributive, "each" (e.g. Deut 1:23, "a man from each tribe)
It will be seen, then, that range of Hebrew ʾeḥād is broader than the English numeral "one", and that "first" is one (!) of its meanings.
For two reasons it is unlikely (almost certainly not) to be taken as "first" in Daniel 10:3, however:
- the use of ʾeḥād as the ordinal "first" in Hebrew is almost exclusively restricted to contexts of date formulae (HALOT lists seven examples under this heading: all of them are in dates of some kind, be it connected to day, month, or year); and
- the formulation here is ʾeḥād in the "construct" (a kind of genetive relationship, so the form is ʾaḥad) followed by a definite noun with the article.
On the latter point, this construction occurs 18 times in the Hebrew Bible (Gen. 21:15; 22:2; 26:10; 37:20; Jdg. 16:7, 11; 19:13; 1 Sam. 26:15; 2 Sam. 2:18; 6:20; 13:13; 17:9, 12; 2 Ki. 2:16; Ps. 82:7; Ezek. 45:7; 48:8; Dan. 10:13), and in every one (!) of those texts, it has the sense of "a single example out of a larger group".
This, then, is the meaning to be understood in Daniel 10:3, and the various English translations tend to reflect the "among-ness" of Michael's status; so, for example, as the Tanakh (JPS ) nicely puts it, "...now Michael, a prince of the first rank, has come to my aid...".
(2) The status of Michael
The "source" cited by the OP has the whiff of anachronism about it. Within the horizon of the Hebrew Bible, Michael is mentioned only in Daniel 10:13, 21; and 12:3. That's it. Gabriel is mentioned only in Daniel 8:16 and 9:21. But Raphael, Uriel, Phanuel, or any of the other archangels are not mentioned at all.
The development of the hierarchy among these angelic beings was largely post-biblical (i.e., follows the period in which the books of the Hebrew Bible were written). The first flowering of the more developed angelology can be observed in the book of Enoch, in which most of the "named" archangels can be found. More about this can be found in the "Michael מיכאל" article from the 1906 Jewish Encyclopedia.2
It is of interest to note, all the same, that even Rashi is not inclined to differentiate Michael in Daniel 10:3. In the Eastern Christian tradition's understanding, by the 4th century AD Michael is ascribed the "chief" role (see the paragraph "Regarding his rank..." at the previous link).
At the time of the book of Daniel, however (and what "time" precisely that is need not concern us here, and is a different question), clearly that more stratified "angelology" was not yet in place. The characterization of Michael as among the foremost of the heavenly beings (who was also styled "prince", Dan 10:13, 21) is surely appropriate.
- These are the main headings from L. Koehler, W. Baumgartner and J. J. Stamm (eds), Hebrew and Aramaic Lexicon of the Old Testament = (HALOT), q.v. אֶחָד; the same headings are represented in the venerable and still useful Brown, Driver, and Briggs, A Hebrew and English Lexicon of the Old Testament (OUP, 1906), pp. 25-26. The headings map this way:
HALOT = BDB
1 = 1
2 = 6
3 = 3+4
4 = 7
5 = 2
See also the אֶחָד entry in TDOT, vol. 1, pp. 193ff. For the more typical ordinal "first" (usually spelled with ʾ, רִאשׁוֹן riʾšôn) see also BDB, pp. 911-12. This term occurs in the phrase characterizing the group of heavenly beings to which Michael belongs: הַשָּׂרִים הָרִאשֹׁנִים = "chief [or foremost] princes".
- R.H. Charles, a leading scholar of apocalyptic literature at the turn of the 20th century, has an article on Michael in James Hastings (ed.), Dictionary of the Bible (T & T Clark/Scribners, 1911) vol. 3, pp. 362-3.