Your question has two possible answers:
Yes, he is a time traveler.
No, he's not a time traveler but instead isn't telling the truth.
You will get a comments like Cearalcast's comment below that will tell you that this doesn't matter, and that Jay's story basically never changes, which is a falsehood in and of itself. It matters tremendously because it creates a scenario where anyone trying to follow Jay's various accounts are placed in the position of either:
If a person following Jay's accounts of 1/13/1999 choose 1 or 2 from above, then it shows that only confirmation bias is being used as the basis of their deductive reasoning. If you cannot accept the fact that someone who lies is not a reliable witness for any account of the events, then you are saying that you are comfortable with people being convicted on the basis of lies. There is no legally or ethically justifiable reason that person can have for telling lies when incriminating someone else, especially when they have been give eight separate chances to tell the entire truth. Jay had provided reasons for his lies, but his interests are not the prosecution's interests, and his accounts of 1/13/1999 should have NEVER been introduced as evidence, and he should have NEVER been called as a witness for the prosecution. The very act of calling him as a witness was unethical, as it immediately put the jury in the position of being forced to consider which lies were more reasonable than others.
My last two cents- How many people commenting would accept either themselves or their loved one being convicted on the statements and testimony of a liar? Additionally, how many people commenting would find it reasonable for a jury to choose to believe all lies without question of a prosecution's witness, or to pick and choose which lies of a prosecution's witness are believable if they or their loved one was on trial?