So, as with many things, the confusing is based on whether you're using the term in a NATURAL sense, or a strict theological sense.
You can distinguish a Christian, for instance, from a Muslim or a Jew, on a NATURAL level, where it's based on a material set of belief (even if the supernatural motive is lacking) consisting of at least belief in the Holy Trinity and the Incarnation
Theologically, if you use the term to refer to those who are Members of Christ's Body, then it's only Catholics who are Christian.
I think that the "Baptism" distinction isn't even in play, so it's a false criterion that someone injected.
Natural: set of (material / objective) beliefts in at least the Holy Trinity and Incarnation
Theological: members of Christ's Body (aka Catholics)
Now, Catechumens are always considered "borderline", with one foot in and one foot out of the Church, or, as St. Robert Bellarmine made a metaphor, "in the vestibule" of the Church. When people became Catechumens in the early Church, they were signed with the sign of the Cross and were in fact permitted to call themselves Christians. Among Christians, then, you had "the Faithful" (baptized + profess true faith = fully Catholic) vs. "the Catechumens" (those in process of becoming Catholics, professing at least naturally the Catholic faith, with something called a fides initialis, a natural analogue to Catholic Faith).
Baptism by itself doesn't make someone a Christian. Let's say you were baptized as an infant but then raised an atheist, and never once in your life professed any belief in Christ, the Holy Trinity, or anything else. Is that person a "Christian" in any sense? No. Baptism by itself doesn't cut it.
I find it unfortunate how few of the clergy actually apply what they learn or are supposed to learn, in Logic, Philosphy, and Theology class.