Two organizations, the MJAA and UMJC, coined that term and they define it in complimentary statements on their websites. There are a number of other organizations whose behavior or reputation is sometimes confused with Messianic Judaism (such as J4J and MIA) but these are almost always polite about not calling themselves Messianic Jewish.
Unfortunately, a more fundamental phrase, also coined by the MJAA and UMCJ, is very rarely used: the Messianic Vision.
In the early days of the MJAA this phrase was often used. It referred not to what being a Jewish disciple of Yeshua looked like, but a specific agenda God was working out worldwide.
One reason P'nei Adonai was successful as a congregation from April of 2004 through August of 2007 was that it stayed focus on the Messianic Vision. For example, when I returned from an MJAA conference I would help maintain a sense of God's larger agenda by reporting to congregants about what new things God was doing as part of the big picture, and what old things were done and ended.
(This focus on the Messianic Vision was also the main reason P'nei Adonai was always a small congregation. People wanting to follow a different agenda would quickly see that the congregation understood the Messianic Vision and was not open to being diverted to a different agenda. These people were almost always nice people. A few of them had a genuine God-given agenda. We would enjoy whenever any of them joined us to celebrate the annual holy days. But on a week-to-week level the agendas were clear and different, so the congregation stayed small.)
Even within an official MJAA or UMJC Messianic Jewish congregation not all congregants will be called to the Messianic Vision. Similarly, a person need not be part of a Messianic Jewish congregation to be called to and participating in the Messianic Vision. But a Messianic Jewish congregation that loses its focus on the Messianic Vision soon becomes unable to answer the question "What are we about besides Jews and Gentiles observing Torah together?"
The Messianic Vision never claims to be everything God is doing. Far from it! God is using many plans and agendas to mature his people and prepare the future. The Messianic Vision is only one of many ways that God shows to heaven and earth how his ways and wisdom are the best things even in a broken and sinful world; see Ephesians 3:10:
...through the assembly there might be made known now to the principalities and the authorities in the heavenly places the manifold wisdom of God...No authoritative definition for the Messianic Vision exists as we start 2009. Neither the MJAA nor the UMJC even define it on their websites anymore.
I can no longer describe it with its current details. Because of a new baby in the home I have not been able to leave my family to attend any MJAA conferences since January of 2008, so I am largely ignorant of what new things God is doing as part of the Messianic Vision. But I remain aware of and called to its fundamentals.
Here are the six foundational parts of the Messianic Vision that do not change.
(Perhaps you will recognize them as a visible part of what God is doing where you worship, even if it is not a Messianic Jewish synagogue. If so, please let me know! I always love to hear stories about what the Messianic Vision currently looks like in different parts of the country and world.)
1. God is calling many of Yeshua's Jewish disciples to participate in synagogues
For about 100 years, from the middle of the 1800s to the 1960s, God worked through a movement called "Hebrew Christians" to help Jewish followers of Yeshua be aware of their Jewish identity and have a Jewish home while also participating in a church.
Since the 1960s many (but not all) Jewish followers of Yeshua have a calling to leave churches and participate in synagogue life. While synagogue attendance rates have been decreasing in most branches of Judaism, they have been rising for Jewish followers of Yeshua, who may try to participate in a local Rabbinic synagogue but usually are forced to form their own synagogues when Rabbinic Judaism does not welcome them.
2. God is starting to restore the first-century roots of discipleship to Yeshua's disciples
To be a disciple as scripture uses that term is a big commitment. We should try to make our lifestyle like Yeshua's. Thus we should study his methods of worshipping, his worldview, and his use of scripture. We should memorize his parables and sayings. We should think twice before differing from his habits and manner of lifestyle.
Since the 1960s there has been a great and continuing increase in more authentic discipleship. As examples, many Gentile churches of diverse denominations are studying Yeshua's first-century identity, learning about the holy days Yeshua observed and hosting Passover seders, and doing worship more like Yeshua did. In general people are still very far from authentic discipleship, but they are interested and getting closer.
3. God is restoring the Messiah's Jewish identity
There is nothing new about scholarly agreement that Yeshua was Jewish. But for centuries Yeshua's followers treated his Jewish identity as something he shed at his ressurection as if it were a set of odd customs he put up with while on earth but "got over" once back in heaven.
Since the mid-1900s, Yeshua's followers have increasingly realized that he still is Jewish: he is the Lion of Judah even in Revelation. This is affecting music and visual art as well as theology and discipleship.
4. God is calling Jews and Gentiles to worship together
For centuries people of Jewish and Gentile ancestry have been wary and distrustful of each other, especially at a house of worship.
Since the 1960s, to a degree beyond what interfaith marriages explain, many synagogues see some Gentiles in their pews and many churches see some Jews in their pews. Also synagogues and churches are increasingly cooperative in interfaith work.
5. God is refining his people in preparation for the Messiah reigning on earth
In synagogues and churches with many Jewish followers of Yeshua the congregants are increasingly finding it awkward to be a bench-warmer. God is actively refining and maturing his people. Those who only want to experience something comfortable each weekend are quickly dissatisfied with congregations in which God is requiring more and depart. Being refined by God (or part of how God refines others) is not easy but it is rewarding.
Part of this refining happens when God challenges individuals. Back in October I wrote about a pattern that could be seen in what God was doing with Sar Shalom.
Part of this refining happens when God uses an entire congregation to challenge another group. I recently wrote about an example of this.
Some non-congregational religious organizations, such as Promise Keepers, have also seen that God tends to do extra refining where Jewish followers of Yeshua gather.
6. God is affirming the state of Israel's right to exist
Finally, God has future plans for not only what scripture calls the faithful remnant of the Jewish people but for all the people of God. Scripture has many prophecies about a great Jewish revival from worldliness to holiness shortly before Messiah reigns on earth. The current nation of Israel, despite the errors and corruption of its leaders, has a right to exist, is blessed by God, and is also part of scriptural prophecy.
To conclude, please keep in mind that the details of the Messianic Vision evolve and change. For example, in the past two years the Joseph Project has become increasing influential in Israel and the TOS March of Life increasing influential in Germany.