It is true that they were Jews. It is true they believed Jesus of Nazareth was the Messiah.
However, they never called themselves "messianic Jews." Others called them "Christians". Throughout the New Testament, Paul referred to them as believers, brothers, etc. Not once did he say "Messianics."
That is a far more recent name. It's a name I have no problem with in terms of a group providing itself a name that also offers an explanation of its positions. There is much about the Messianics that I've run across that is admirable. IMHO, they need to work a bit more on how they explain their Trinitarian views -- because they do appear Trinitarian to me -- but they get there by a circuitous biblical logic that makes the head spin.
. Throughout the New Testament, Paul referred to them as believers, brothers, etc. Not once did he say "Messianics."
Trinitarian views: Sorry they were not called believers they were called
shalom b'SHEM Yah'shua HaMashiach
Disciples of Yah'shua HaMashiach
by Paul.
See Acts 11:19-26 Paying particular attention to v 26 29 Jesus answered, “The foremost is,
‘Hear, O Israel! YHvH our God is one YHvH;
Actually, the Jewish followers were more commonly called the Nazarines or Natzerim (the "Branches," as in, "I am the vine and you are the branches), being followers of Yeshua HaNatzeret (Nazareth, "Branch-town"), who is the prophesyed Branch of David. They were also known as the followers of The Way.
The term "Christian" is only used a couple of places in the NT, and in both cases, it's a term that the movement's enemies used of it. Interestingly, putting "-ian" at the end of a word in Latin makes it a political term, like "Republican." Some of the expositors (and I don't have the quotes on hand, being at work) point out that the term is somewhat disdainful.
As with with Cross itself, the gift of Christianity is taking things its enemies mean as a mark of shame and turning it into a badge of honor.
Nevertheless, the term Christian has taken on a meaning that I don't believe the Apostles or Paul would have completely identified with. Yes, a Christian is a follower of Christ, but implicit in the term is someone who goes to a church on Sunday, celebrates Christmas and Easter, happily chows down on pulled pork at the barbecue, etc. In other words, a Gentile, following Christ and worshipping God in a distinctly Gentile fashion.
The Apostles and that first generation of Yeshua's Jewish followers continued worship in the Temple right up until it was destroyed, went to synagogue on the Sabbath, kept the Feasts of the Lord given to Israel, kept kosher, took Nazrite vows and fulfilled them, and even maintained their membership in their original sects of Judaism (ala Paul still being able to say under oath twenty years after his Damascus Road experience that he was still a Pharisee and the son/disciple of the Pharisees).
If one of the Apostles walked into a church today, the Christians in that church would identify him as a Messianic Jew.
Ergo, based on what the words actually mean today and how they use them, Uriel is right that everyone would identify the Apostles as Messianic Jews based on their identity and practice.
Shalom.