You are, of course, wrong. Cyprian and Firmilian were not in communion with Rome but they referred to themselves as Catholic and rightly so. You need to broaden your horizon beyond RC apologia.
You can't possibly be reading the same Cyprian that I am:
Cyprian of Carthage
"The Lord says to Peter: I say to you, he says, that you are Peter, and upon this rock I will build my Church, and the gates of hell will not overcome it. And to you I will give the keys of the kingdom of heaven; and whatever things you bind on earth shall be bound also in heaven, and whatever you loose on earth, they shall be loosed also in heaven [Matt. 16:1819]). ... On him [Peter] he builds the Church, and to him he gives the command to feed the sheep [John 21:17], and although he assigns a like power to all the apostles, yet he founded a single chair [
cathedra], and he established by his own authority a source and an intrinsic reason for that unity. Indeed, the others were also what Peter was [i.e., apostles], but a primacy is given to Peter, whereby it is made clear that there is but one Church and one chair. So too, all [the apostles] are shepherds, and the flock is shown to be one, fed by all the apostles in single-minded accord. If someone does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he [should] desert the chair of Peter upon whom the Church was built, can he still be confident that he is in the Church?" (
The Unity of the Catholic Church 4; 1st edition [A.D. 251]).
"Cyprian to [Pope] Cornelius, his brother. Greeting. . . . We decided to send and are sending a letter to you from all throughout the province [where I am] so that all our colleagues might give their decided approval and support to you and to your communion, that is, to both the unity and the charity of the Catholic Church" (
Letters 48:1, 3 [A.D. 253]).
"Cyprian to Antonian, his brother. Greeting ... You wrote ... that I should forward a copy of the same letter to our colleague [Pope] Cornelius, so that, laying aside all anxiety, he might at once know that you held communion with him, that is, with the Catholic Church" (ibid., 55[52]:1).
"Cornelius was made bishop by the decision of God and of his Christ, by the testimony of almost all the clergy, by the applause of the people then present, by the college of venerable priests and good men ... when the place of Fabian, which is the place of Peter, the dignity of the sacerdotal chair, was vacant. Since it has been occupied both at the will of God and with the ratified consent of all of us, whoever now wishes to become bishop must do so outside [the Church]. For he cannot have ecclesiastical rank who does not hold to the unity of the Church" (ibid., 55[52]:8).
"With a false bishop appointed for themselves by heretics, they dare even to set sail and carry letters from schismatics and blasphemers to the chair of Peter and to the principal church [at Rome], in which sacerdotal unity has its source" (ibid., 59:14).