‘Catholic’, in modern English usage, is a generic term that does not exclusively belong to churches which are faithful to the Bishop of Rome. Any number of Reformational denominations recite the Apostle’s and/or Nicene Creeds, including the part that says “I believe in the holy catholic Church.” Anglicans often prefer to speak of themselves as ‘catholic’ with a small ‘c’. Many Episcopalians don’t see themselves as Protestants in the same sense that Lutheran and historic Reformed denominations see themselves as ‘Protestants’. They feel they are part of the continuing catholic Church as it existed before the Bishop of Rome attempted to usurp political influence. Episcopalians often don’t feel they are reforming the Church of doctrinal, liturgical, or clerical excesses, but rather are functioning as an independent episcopal jurisdiction as the Enlgish Church operated prior to the 1200’s.
Many second- and third-generation Protestant denominations are not very comfortable with the term ‘Protestant’ because they were not born in protest against the Roman Catholic Church. They feel themselves removed from those fights and think they should be known more for the principles they stand for rather than the things they, or their forefathers, were against. Pentecostals, for example, want to be recognized as People of Pentecost, people who believe in the continuing miraculous gifts of the Holy Ghost.
The denomination I was raised in, the independent Christian Churches/Disciples of Christ, see themselves as “Christians only, though not the only Christians”, and so they rejected the term ‘Protestant’. Many prefer the term '“restorationist”, although that term has also been appropriated by the Latter-Day Saints and their splinter groups. (Both the Mormons and the early founders of the Christian Churches/Disciples of Christ drew heavily on an earlier movement known as ‘restorationism’, or the ‘Christian Connection’, which held that the Christian Church was not in need of a Reformation so much as a restoration. How that restoration could take place was hotly debated-. Alexander and Thomas Campbell and Barton Stone held that the Church was not lost entirely, merely adrift from it’s aboriginal purity. Therefore they felt restoration could take place via a careful adherence to Christianity as seen in the Acts of the Apostles. Other restorationists felt that the Church of Christ was gone utterly and that some sort of direct act of God would be needful for the church to be properly restored–making such persons ripe for the claims of Joseph Smith).
As has been noted, non-denominational churches take their cues from the same impulse that sparked the “Christian Connection”–they want to emphasize that they are “Christians only”, rather than focus upon what they may be ‘against’. Ultimately that’s the biggest issue I think you’ll find that many non-Catholics have with the term ‘Protestant’–they often feel that the term ‘Protestant’ does not adequately describe what they support, advocate, and believe in. They do not want to be known merely for what they are NOT.
