Apologetics came about early in the history of the Church and developed over the centuries. To categorise this vast historical record there is a need to know the history of its development, and who the Christians were who became known as the Apologists, and the subjects they both defended and promoted as necessary for Christians to believe. Religious Apologetics is a discipline of defending religious doctrines through systematic reasoning. In modern times, Apologetics is often thought of as theological exchanges between one group holding to a particular religious belief, as compared with a different group's belief. Apologetics and systematic theology are closely linked.
This question, in asking if Apologetics is practiced as a single discipline, has the answer, "Yes," which is the same answer to whether theology can be practiced as a single discipline. The trouble is, the scope of both is vast and they often intermingle. You asked if there is "a special class of Apologetics which focuses on the Truth of the Bible, for example, and doesn’t involve arguments for an afterlife." The problem here is that the theology of both those topics are bound together, and before any Christian can be an Apologist on either topic, they have to know the theology of both. Only then might they go on to be a successful Apologist for both topics, and they could well treat each topic separately. But that would depend on what contrary beliefs they were defending the topics against. To be an Apologist dealing with contrary Buddhist beliefs on authoritative scripture, for example, might require showing the incompatibility of their respective beliefs in the afterlife, which would be shown up by comparing their respective scriptures. Likewise with an Apologist dealing with contrary Muslim beliefs. Even within the ranks of different Christian groups, there is usually a related clash between more than one doctrine.
There are lists of the Apologists who wrote defenses of the Christian faith and who also wrote theologies. They were called "teachers of the church" more often than they were called "Apologists". The simple point is that nobody could become a Christian Apologist without first having a deep grasp of a whole range of theological topics. And this is where another point you raised is crucially important.
"Apologetics in the Word won’t do much for a person who doesn’t yet accept the person of Jesus." So true! This demonstrates how it's possible to put the cart before the horse, theologically speaking. In fact, to be an Apologist seeking to uphold the integrity of scripture without having yet developed belief in what the scriptures say as to who this Jesus is, is an exercise in futility. The New Testament scriptures start by showing just who this holy one who has been born to the virgin Mary really is - "born this day, ...a saviour, which is Christ the Lord" (Luke 2:11); "he shall save his people from their sins... Where is he that is born King of the Jews?" (Mat. 1:21 v& 2:2). The apostle Paul starts his gospel message with pointing to the person of Christ, before detailing what this person did. In Romans 1:1-4, the evangel does not start with the work of Christ, but with the Worker. It commences with whom he is; it declares his person. What's the use trying to proclaim the gospel to a person without having first pointed them to the glorious personage of the gospel?
Genuine conversion to Christ, the resurrected Son of God, opens up the theology related to Christian belief and then a person is enabled to speak properly, either as a teacher in the congregation, or as a public Apologist. Yet it is certainly impossible for a non-converted person to succeed at being a Christian Apologist, no matter how erudite or how full of head-knowledge they are about theology.
What this boils down to, with regard to the question, is that the formally recognised categories of Apologetics avails no-one in and of itself. Recognise who any given Apologist is; a genuinely converted Christian, or someone just out to spread empty knowledge about a subject? All the subjects within Apologetics must first be subjected to the authority of the One being spoken about, for all scripture is of Christ, the living Word of God, and all the Holy Spirit's activity is to glorify Christ, who testifies of Christ, who receives of Christ, and shows that unto those being saved (John 15:26 & 16:14).
It is who you believe that matters here. Is the Apologist able to say, with Paul, "I know whom I have believed, and that he is able to keep that which I've committed unto him unto that day"? If so, then go on to consider his defense of whatever Christian doctrine he speaks of. Otherwise, all his rhetoric and head-knowledge will count for nothing.