This is partially correct, the Church has always believed itself, and still does, to be the One True Church, this is a popular myth that somehow this belief has changed, it has not, but as you say, the Church does believe that it is the fountain of all Graces, flowing upon the earth, and through the mystery of this Grace, others can be saved. That is the mission of the True Church, to be the mediator of Gods Grace for the salvation of souls. In the Creed we say, “We believe in the ONE, HOLY AND APOSTOLIC CHURCH”. I’ve also had experiences with newer Protestant denominations re-evangelizing ‘good’ Catholics believing them not to be Christians at all, following a cult. The Roman Catholic church is Cultic in it’s practice, but not a cult in the same vain as JW’s.
The Catholic church is not a Cult, it is arguably a Sect, but even so, not in the negative context of the word. Christianity was a Sect itself, in fact most religions start off as Sects of other religions, in the case of Christianity, this was of cause Judaism as Jesus was Jewish.
Christianity, no thanks to Luther, is now a religion which is Sectarian itself in its nature. It is now not a unified Religion in its entire belief system even though some protestant “pastors” would have you think otherwise. People who are in a Sect of a religion believe the common principles to follow that particular religion, but also believe something extra on top of that which does not alter the interpretation of the previous beliefs, this is. A cult is a small religion entirely of itself,. i.e not sectarian in nature. For example, Scientology. It is a Religion(as all cults are), but it’s a cult too. And Jehovahs witnesses are also the same because they deny the basic tenant of Christianity that Christ was the Son of God. They are Christians by pure definition of the word, and they are not part of the christian religion themselves in the basic tenant sense, only in Definition(this also covers Mormons).
Judaism and Islam are not sectarian by nature, but they are also not cultic because they have large numbers(which separates a cult from a Religion), Generally the beliefs of these two are pretty streamlined, even though there can exist sects within.
If one looks at the basic combining beliefs of the entire christian world, you would certainly understand that the Catholics are in fact also Christians.
They believe in Jesus being the son of god
They believe in his Crucifixion(for our trangressions), Resurrection and the subsequent resurrection of ourselves at the end of time.
They believe in the Holy Trinity
They believe in the bible being authored(or at least inspired by) by God.
I think these are quite representational of the combined basic beliefs of the Christian faith, and because they are held by the Catholics, they are Christians in the biggest of the many thousands of Christian Sects(or denominations if you will).