Definition of a Catholic

[Genesis315]

I would define a Catholic as a member of the Catholic Church (or in equivalent words used by the Church, one fully incorporated into the Church). So what does it mean to be a member of the Church?

This is how Vatican II defines a Catholic:
Lumen Gentium 14:

They are fully incorporated in the society of the Church who, possessing the Spirit of Christ accept her entire system and all the means of salvation given to her, and are united with her as part of her visible bodily structure and through her with Christ, who rules her through the Supreme Pontiff and the bishops. The bonds which bind men to the Church in a visible way are profession of faith, the sacraments, and ecclesiastical government and communion.

Pope Pius XII taught it as the same thing:
Mystici Corpois 22:

1. Actually only those are to be included as members of the Church who have been baptized and profess the true faith, and who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed. “For in one spirit” says the Apostle, “were we all baptized into one Body, whether Jews or Gentiles, whether bond or free.”[17] As therefore in the true Christian community there is only one Body, one Spirit, one Lord, and one Baptism, so there can be only one faith.[18] And therefore, if a man refuse to hear the Church, let him be considered - so the Lord commands - as a heathen and a publican. [19] It follows that those who are divided in faith or government cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit.

As an aside, unity of faith does not mean everyone must know every jot and tittle of Catholic doctrine. Just being ignorant or wrong won’t separate one from the Catholic Church, as long as one is not meaning to rebel against the Church. In those cases, as Pope Innovent IV explained, “in that case, the faith of the Church replaces his opinion, though his opinion is false, it is not his faith, but his faith is the faith of the Church.” (Commentaria in quinque libros decretalia, Ad liber I).