In other words, “a being is a man if and only if they are an animal and rational.” The moment a being ceases to think rationally, it ceases to be a man.
You’re misunderstanding Objectivism’s notion of man. In it’s fundamentals, Objectivism has no different conception of man than a good biologist would have. Indeed it has no radical view about who is a man and who is not. However, one does not judge this closeness of conception by an analysis of definitions. Rather, one considers all the actual existing entities (particular animals, in this case) that one would term “man”. Objectivism offers no radical new sub-set; it does not say: all these animals that you referred to as men: like Jesus and Marx, guess what … …they aren’t men.
Now, I understand why you may be confused, because you are taking the definition to be the starting point. It is not. The definition is a neat little summarization that describes the millions of entities being subsumed by a concept. Suppose a scientist comes up with a coding scheme for DNA and we get a definition like “Man is a DNA-X456Z-12 animal”. Such a definition is not the starting point for the scientist. Rather he looked at all sorts of men, sought their defining characteristic, and came up with one. While doing this, he might have encountered a few men who have some abnormalities. However, whatever scheme he tried, there were always a few men out of a million that had abnormalities. Is he left in a position where he cannot define man? Some might say yes, he’s stuck. However, Objectivism does not. Definitions serve a specific purpose: they help us tie a concept back to a single (though important) aspect of the existents subsumed by a concept. Definitions act like a reminder about the reality of the existents. With this intent, the scientist would be perfectly correct to use his definition, despite the few exceptions.
Consider this: from a certain perspective, the scientist’s definition is better than the “rational animal” one: i.e. from the perspective of the percentage of abnormal existents included in the concept, but not covered by the definition. Yet, a definition like that is not better from most perspectives: from ones that are not interested in thinking about detailed biology. From a broader and more lay-person perspective, man is an animal (still a biological perspective, but very simple); and he has a brain that has qualities that even a layman understands is significantly different from other animals. Also, like the scientist’s definition, this particular distinguishing characteristics is an important one when one considers all the other characteristics of men. For instance, we could define man as a plane-flying animal, or as a theorem-forming animal, or as a forward-thinking animal, but defining him as a rational animal evokes plane-flying and theorem-forming and forward-thinking. So, we settle on “rational animal”, not as a way to start to exclude things that were in our set of things we call men, but as the best way to succinctly describe that set.
When Objectivism says “man is a rational animal”, what is meant is that man has a certain biological mechanism that allows him to be rational. People who have such a mechanism and do not use it well are also included. This is why Objectivism says Rationality is a virtue: because while all men have the means to be rational (lunatics excluded), they do have a choice as to whether they use their rationality.
Now, if you want argue that Objectivism has worded its definition poorly, and that it ought to say “Man is an animal with the potential to be rational”, you can do so. Someone here might like to argue the other side; but, it’s a side argument that does not change what Objectivism is saying. The whole focus on telling men they ought to be rational implies the potential not to be so. In fact, Rand makes so many arguments about various other theories of philosophy, psychology and economics being irrational. There is no implication that she thought that Kant or Keynes or James Taggart were not men.
I have not addressed the link from rationality to rights, but it was important to get this out of the way first. Check out Trebor’s link, and get “Introduction to Objectivist Epistemology” for more info.