I have not studied this but was directed to a Wikipedia article in my early search on the subject, so this subject is incomplete and posted for comments by some more educated with the history of this. It's my impression that Jews that came to Christ were Christians and would have called themselves such, or perhaps make a distinction by calling themselves Jewish Christians. The pastor at a Calvary Chapel I attended asked those in the congregation who were Jews to raise their hands, and I was stunned by the number of Jews among us.
According to the Wikipedia article it would seem that prior to the 19th century there was no desire of Jewish Christians to embrace old covenant law, and efforts went more toward Bible translation:
https://en.wikipedia.org/wiki/Messianic_Judaism#Pre-19th_centuryThen during the 19th century (wouldn't you know it!):
"In Eastern Europe, Joseph Rabinowitz established a Hebrew Christian mission and congregation called "Israelites of the New Covenant" in Kishinev, Ukraine in 1884.[45][46][47][48][49][50] Rabinowitz was supported from overseas by the Christian Hebraist Franz Delitzsch, translator of the first modern Hebrew translation of the New Testament.[51] In 1865, Rabinowitz
created a sample order of worship for Sabbath morning service based on a mixture of Jewish and Christian elements. Mark John Levy pressed the Church of England to allow members to embrace Jewish customs.[47]
In the United States, a congregation of Jewish converts to Christianity was established in New York City in 1885.[52] In the 1890s, immigrant Jewish converts to Christianity worshiped at the Methodist "Hope of Israel" mission on New York’s Lower East Side while retaining some Jewish rites and customs.[53]
In 1895, the 9th edition of Hope of Israel's Our Hope magazine carried the subtitle "A Monthly Devoted to the Study of Prophecy and to Messianic Judaism", the first use of the term "Messianic Judaism".[54][55] Hope of Israel was controversial; other missionary groups accused its members of being Judaizers, and one of the two editors of Our Hope magazine, Arno C. Gaebelein, eventually repudiated his views and, as a result, was able to become a leader in the mainstream Christian evangelical movement.[54]"
https://en.wikipedia.org/wiki/Messianic_Judaism#Pre-19th_century"The modern day Messianic Jewish movement is considered by many to have begun in the United States in the 1960s and maybe even more specifically in 1967.[60][61][62][63]"
"The Six-Day war (The Arab-Israeli war of June 5 to June 10, 1967) brought Jerusalem back under Jewish control.[65][better source needed] This was seen as a fulfillment of Jesus’ prophecy: “Jerusalem will be trampled on by the Gentiles until the times of the Gentiles are fulfilled.â€(Luke 21:24 TLV)[66][67][68][69]"
"Transformation to a Jewish Identity and Culture
Prior to this time Jewish believers, had in most cases, assimilated into gentile Christianity, losing their Jewish identity and not passing on their Jewish heritage to their children. Now there seemed to be a greater desire and urgency among the young believers to maintain not only their individual Jewish identity[75] but to celebrate their Jewish traditions and culture with their families and Jewish friends.[76] Understanding Jesus within his historical Jewish context was of great importance as well as seeing Jesus within modern day Judaism especially in the observance of the Shabbat and the celebration of the festivals.[77] This conflict of identity as a Hebrew-Christian and passion for their Jewish identity was summed up by Paul Liberman where he writes about the founding of Beth Messiah Congregation.
'Almost as soon as Beth Messiah began to hold meetings on May 18, 1973, I realized I would be required to “explain†our purpose to the outside Jewish community. I needed to be able to honestly profess that we were not an appendage, subsidiary, affiliate, or offshoot of any Christian (Gentile) denomination or ministry. We were a Jewish congregation and always would be. We absolutely were not a “church.†For this reason, if we identified as, “Beth Messiah, a Messianic Synagogue†immediately followed by, “affiliated with the Hebrew Christian Alliance of America,†it would seem an inherent contradiction in our stated goal. How could you have a “Christian synagogue?†… If this fledgling thing –Beth Messiah—was to have a future, it couldn’t be under the banner—Hebrew Christian Alliance?â€[78]'
This idea of a Jewish identity and more specifically a Messianic Jewish identity was the main purpose of the effort to change the name of the Hebrew Christian Alliance of America (HCAA) to Messianic Jewish Alliance of America (MJAA).[79][80][78] Dan Juster comments on this by saying that “Hebrew Christianity, at times, saw Jewishness as merely an ethnic identity, whereas Messianic Judaism saw its Jewish life and identity as a continued call of God.â€[81] David Rausch further explains that “(t)he name change, however, signified far more than a semantical expression—it represented an evolution in the thought processes and religious and philosophical outlook toward a more fervent expression of Jewish identity.â€[82]"
https://en.wikipedia.org/wiki/Messianic_Judaism#The_Modern-Day_Messianic_Jewish_Movement.2C_1960s_to_present