Hajj Muhammad Legenhausen teaches at the Imam Khomieni Education and Research Institute in Qom, Iran. His blog can be accessed at httppeacethroughunderstanding.blogspot.com

Of course there are jurisprudential answers to the question of who is a Muslim, but they only provide some criterion that can be used to settle issues in Islamic law, e.g., inheritance, without any regard for what one's true beliefs or allegiances are.
Even when we narrow down being a specific sort of Muslim, (Shi'ite, 12 Imami, for example), there is still always a lot of contradictory things that fit the label. Some are liberal, some conservative, some socialist, some free market proponents, etc.
Of course there are jurisprudential answers to the question of who is a Muslim, but they only provide some criterion that can be used to settle issues in Islamic law, e.g., inheritance, without any regard for what one's true beliefs or allegiances are. So, the issue of how to define what it means to be a Muslim needs to be considered in terms of various purposes or goals that we might have in making the distinction.
The distinction between general and specific Islam will apply to some, but not necessarily all of these ways of being a Muslim. We should at least not confuse the following

1. Definition of being a Muslim in jurisprudence.
2. Affiliation with the Muslim community. This can have at least two forms (a) acceptance by a community of Muslims of which one is considered a member, or (b) one's own identification with a specific group or with the ummah generally. In (a), affiliation is determined by the attitudes of others; in (b) it is determined by one's own attitudes. Affiliation sometimes takes the form of allegiance with political significance during the early years of Islam, and continued with respect to spiritual authority by the Sufis. Even without the outward form of allegiance, affiliation can be constituted by mutual acceptance in a relationship between some authority and follower, where the authority is defined in terms of political leadership, teaching authority (in law, doctrine, exegesis, etc.), or spiritual authority.

3. Holding the values and doctrines taught in Islam. This can take various forms. One might suggest there are diverse forms such as holding the values and doctrines taught by Islam according to (a) the Wahhabis, (b) the Traditionalists, (c) the 'ulama of Qom, (d) Mulla Sadra, (e) the Mevlevis, (f) the awliya’, (g) one's own understanding of what these are to the best of one's abilities.

4. Behaving in accordance with point 3. On this point there was a major controversy in the history of Islam, with the Kharajites holding that anyone who committed a major sin is not a Muslim, and Muji'ites, who held that the grave sinner might be both Muslim and a believer. According to the great Shi'ite theologian Shaikh Mufid, the grave sinner may be a Muslim, but not a believer in the full sense, rather a sinful believer (mu'min fasiq). But the issue here was whether major sin was an obstacle to being a Muslim in the legal sense. One might also hold that Zayd is a Muslim in the legal sense, but not in a behavioral sense, because, e.g., Zayd is a hypocrite.
Now, with regard to general and specific Islam, in jurisprudence, there is no such distinction. Criteria are only given for specific Islam; general Islam is not a legal category. Mu'min is also not a legal category.
With regard to affiliation, we could speak of specific and general varieties, although I have not seen anyone doing this. But theoretically, we could consider affiliation with those who are Muslims in the general sense of submitting to God and seeking truth in all sincerity. And the same could be extended to 3 and 4.
In the end, the labels don't matter much. What does matter, however, is what has traditionally been called knowing one's Imam, as well as living in a community of faith, drawing strength from that community, and accepting its ordinances, enjoining the good, and forbidding the bad. (This, however, is often abused, so finding a community in which this is properly practiced may be particularly difficult.)
For the Shi'a, living in community is focused on guidance by the living Imam. In this period of Major Occultation, the issue of how to be guided by the living Imam becomes fraught. On a political level, it is understood by most of the scholars today in terms of Wilayat al-Faqih (Guardianship of the Jurists), and on a jurisprudential level, it is understood in terms of Marjaiyyat. But there are many other levels that I dare say are more important than the political and jurisprudential.
Some Shia Sufis understand guidance on a spiritual level in terms of relationship with a Qutb. But for most of the Shia, matters of doctrine, spirituality, ethics, philosophy of life, etc., are things one has to find one's way through means of personal study and discussion and other activities in which one individually or in company with others an attempt is made to follow the guidance of the living Imam as best we can under the circumstances. This has implications for how we should live. Our following the Imam has to have visible manifestations in the way we live and in the manifestation of virtue. This is the main message of such books as Fada'il al-Shi'a and Sifat al-Shi'a of Shaikh as-Saduq.