Interesting you specifically would point out in your "example" that Socrates trained the guy who trained Aristotle. Unless of course now you don't believe (or in TRUT terms "never said") Aristotle was real.
1. It was an exegetical example to demonstrate how writing, specifically how syllogistic writing, ought to proceed. I could have used unicorns, as well. Or, if you like, here's an example of a syllogism: P1: If Jesus was the son of God, Jesus was divine; P2: Jesus was the son of God; C: Jesus was divine. Yet, I do not believe that Jesus existed, but I not only can offer that as an example, I have offered that exact example to students to whom I have taught logic.
2. To posit something is to declare it as true. Amazingly, I do believe that Socrates did exist, yet I don't declare, "Socrates existed" as true. Why? Because I do not know if he existed. I cannot confidently assert that my belief is true, thus I cannot posit that Socrates existed. I can assume; I can hypothesize; I can guess. But, I do not know, and thus I do not posit.
3. Herein lies the distinction: Christians don't merely assert that they believe Jesus existed, they posit his existence as truth. When someone questions this truth-claim, then they say that the burden of proof is on the questioner. No, it is not. I someone declares they know something to be true, then they are claiming that they possess a justified, true belief; not merely a belief; not merely a justified belief. As such, the burden is on them to present the justification, if they are claiming knowledge of truth.
4. I have no argument for Socrates's existence. But, I don't really care if he existed or not; i.e., I don't care if my belief is false. Because, I have nothing staked on the belief. If Socrates never lived; if Plato never lived; if Aristotle never lived; I would not be in the least bit concerned. If the works bearing the names of Plato and Aristotle were not actually written 2,500 years ago, but were in fact written in the late 1970s, just before I was born, this would not bother me one bit. Why? Because it is not important to me who wrote those works; it is important what is written in those works. The arguments are great arguments; the questions are great questions. If it turned out that a monkey pounding random keys on a typewriter was able to spit out a random string that just happened to be The Republic, great!
5. The problem is that Christians do have a stake in whether or not Jesus existed. In fact, they think they stake their entire life and after-life on this claim. Thus, they don't question whether the belief is true or not; further, they immediately ask for proofs of anyone who does question it. And, that is ass backwards.