For a while now, I've been calling most American churches by the general description of Post-Reformation churches. This is not how most of my fellow Catholics described them, nor is it how members of most American churches would describe themselves. I certainly didn't describe myself that way when I was attending Pentecostal churches, nor did anyone I knew.
Catholics generally describe the members of all American churches as Prodistants and label their churches Protestant churches, and the members of those churches generally use the term Christian without for themselves any additional qualifier beyond the name of a denomination, a claim to be the one true Christian church, or a claim to be non-denominational.
My fundamental reason for calling these churches post-Reformation churches is that I believe in the importance of using precise and accurate terminology to the degree that it's possible. When I use the term Protestant, I'm specifically referring to churches whose teachings are or contain a primary element of protest against Catholic teaching or the corrupt Church officials in pre-Reformation Europe.
The Protestant Reformation was the seminal event of the Great Western schism, but it was far from the end of it. The Protestant Reformation ended in the 1700s, but the schisms among the churches that had sprung up among Christians after the Reformation continued unabated for the next couple of centuries. Many of these churches were not operating in protest against Catholic teaching or corrupt Catholic officials at all; they were breaking away from the Protestant churches or from other Post-Reformation churches over internal doctrinal, pastoral, or social disputes.
Because it is very difficult to use a blanket term for these churches that were formed over disputes among Protestant church members or other Post-Reformation church members to describe such a wide array of churches who have been established based on thousands of minor variations, I decided to use Post-Reformation churches as a catch-all term while acknowledging that it has obvious limits.
Despite those limits, I find it very useful because it makes it clear that the theology, worship, and praxis of these churches is influenced by the Reformation while not over-emphasizing their connection to Reformation figures like Martin Luther, Huldrych Zwingli, or John Calvin. In many cases, these Post-Reformation Christians have not read the works of Luther, Zwingli, or Calvin. Even those who have some basic understanding of Calvin's view of predestination have usually not read the Institutes of the Christian Religion to gain a deeper understanding of Calvin's form of Christianity.
The other reason that I prefer to use Post-Reformation as a descriptor is that it's not prone to be used as a pejorative term like Protestant or Prodistant. In general, I prefer to avoid pejorative terms that might prejudice the reader or listener against a particular point of view, even when I disagree with that point of view, which I now do.
I gave up the Protestant Intuitions long ago myself, but they live on in the minds of many of my fellow virtuous and upstanding citizens, and there's no good reason to go around insulting them or describing them uncharitably, so I'll continue calling them Post-Reformation Christians.
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Note: Above is a picture of Martin Luther's edited Bible translated into German.