First, I am a product of traditional Roman Catholicism. So this peppers my thoughts on this topic. I belonged to groups which truly believed they were the remnant of the Roman Catholic Church because Rome espouses heresy. But even these groups have common understandings which I think is important for Indie groups.
1. Traditionalists are more likely to be especially vigilant about union with Rome. In fact, the Priestly Fraternity of Saint Peter was founded by priests uncomfortable with the perceived schismatic actions of Archbishop Lefebvre. The majority of Roman Catholics prefer to remain on the Barque of Peter, even if they feel the boat is going down the wrong stream. The nuances of jurisdiction may not be apparent to the average Catholic, where as the traditionalist is likely very aware of the consequences. This isn't a condemnation of non-traditionalists, but more is a realization that this is an important topic to a certain section of Catholicism. I also think that this exists less with Anglicans, Lutherans, etc. because their faith tradition does not hinge on a central person as the Vicar of Christ. Few people attending an ACNA parish, for example, are terribly concerned that it isn't under the Archbishop of Canterbury.
2. Traditionalists are apt to make it very clear that they are not under the local diocese. Sometimes painfully so by condemning the local bishop, the Pope, etc. They may have apologists who argue that they are not in formal schism, either because they hold the “true faith” or because they regard themselves as Roman Catholics, but most often they are very clear that they are independent of the local diocese. I find it problematic if an Independent Catholic group does not do the same--not just problematic but misrepresenting, the same as if a traditionalist group did not make this distinction. This is especially true, for me, where there is a language barrier. I have heard horror stories of Latino people attending a parish without the realization that it is not a parish under Rome.
There are theological differences, too, between progressive groups and traditionalist groups which cannot be ignored. Traditionalist groups follow the historic books of the Roman Catholic Church. While one can argue about their obedience and canonical status, they do follow the tenants of Roman Catholicism such as existed prior to Vatican II. The same cannot be said of progressives who call themselves Roman Catholics, especially if they are LGBT affirming, ordain women, etc. My mentioning these things is not to take a side on them, but just to say that there is deviation from what can be considered the faith of the Holy Roman Church.
But just because traditionalist groups follow the traditional faith (and liturgy) does not give them a pass as Roman Catholics. I realize some will disagree with me here, but I believe to be a Roman Catholic one must submit to the Roman Pontiff. Vatican Council I occurred whether one likes it or not, and the Roman Catholic Church is an Ultramontanist institution. While there are those of us who espouse the Old Catholic tradition (which is not solely dependent on Papal Infallibility but also is impacted by issues of local control, election of bishops, etc.) and may even consider ourselves to be closer to the historic version of Catholicism in the West, we realize that we are no longer Roman Catholics. To argue differently after 140 years since Vatican I is fantasy. Thus I would actually place many of these groups which are not subject to the Pope but call themselves Roman Catholics into our movement. Because they are not under the jurisdiction of the Pope, something which has been settled by the majority of Roman Catholics for 140 years, they are outside the Roman Catholic Church.
With these realizations, it doesn't mean that I don't realize the challenges with naming. I was discussing this with a bishop friend: how do we exist in the periphery of Rome and often ministering to Roman Catholics without misrepresenting ourselves. Another bishop friend told me that, for this reason, he is not fond of the word "Catholic" as a church name. And I get his frustration. There are different options--we see this with Orthodox groups like the Armenian Apostolic Church, the Maronite Church which is in union with Rome, etc. But Apostolic in America is traditionally associated with groups in the Black Church tradition, Orthodox has a "whole 'nother" set of issues, etc.
In spite of these naming challenges, I still don't think that using the name Roman Catholic is the way forward. First, it damages our relationship(s) with local Roman clergy. Many of us have been fortunate to have good relationships with local clergy and this puts that in jeopardy. Second, it ignores our Autocephalous Catholic tradition. I remember fondly when a very conservative Roman Catholic said, in response to someone who said my group is just a start up group, "oh no, they have their own history and background." Finally, just because someone uses the name "Roman Catholic" doesn't make it true. The Roman Catholic Women Priests can't claim to the Roman Catholic priests in good standing any more than the groups of Roman Catholic priests who left to marry and are no longer priests in good standing. Or, priests who were removed or left voluntarily for other reasons.
There are possibilities. A group can call itself "in the Catholic tradition" or use any number of phrases as long as they indicate that they are not under Rome or affiliated with the local diocese. People have to have the tools necessary to decide for themselves--if they will attend the parish despite the lack of Roman approbation. And, frankly, many will. But clarity is necessary and ethical. I have heard too many stories of laity who attended a parish that did not distinguish itself and left angry and hurt.
“One might say that to the East the pope was chief bishop because he was orthodox, while to the West he was and always would be orthodox because he was chief bishop.” - Jaroslav Pelikan
