I lecture you on faith, not on religion. Religion is just another human institution we use to try to make sense of the divine. (IMO)
Frankly, I could care less about what specific day Jesus was or was not born on. I could care less about how my belief in the stories of the Bible matches or doesn't match other religions.
For me, Christmas is all about one thing. I believe it was important that Jesus was born (because if he were not born man, he could not die as a man for man on the cross). That is what I believe, and that is what -- and why -- I celebrate.
The actual day is just a bit of human record-keeping. ("History", too, is a human institution.) As a historian, I'm interested in the chronology of the ancient world; but, in this, my faith has to trump my vocation. Because I still believe that history and other tale-telling is limited by the abilities of the tale-tellers. But this is about God, and God is not limited by human political, religious, or scholarly shortcomings. To think we can reduce Him into "what we know" is nothing less than idolatry of ourselves.
That is why I resist the "religion" tag. Religions are in the business of trying to reduce God into something consistent with them. That was true at Nicea, and it was true at Worms, it is true at the Vatican, and it is true in every other denomination that has ever claimed the authority of God for their own rules and practices and limitations and restrictions.
I don't believe Peter was infallible -- for Pete's sake, he denied Christ three times -- and I sure as heck don't believe there's ever been an infallible Pope. Despite my personal admiration for Paul, I don't believe Paul the man was capable of infallibility. (I do believe that Paul-as-inspired-amanuensis was striving to put God's Truth into human words. Despite calling myself a Luther-an, I don't even believe Luther got God 100% correct. (In fact he explicitly denied that he could over and over again. It wasn't "by Scripture alone as I have interpreted it," it was "by Scripture alone." Period.
If it helps you make sense of God and strengthens you in your faith to believe that the exact date (or even a more accurate one) matters, then by all means try to get the history right. But don't let the historical scholarship lead you to rejection of the faith.
That is the problem of the "higher learning." We who study for a living are always risking ourselves under the foot of pride; for we being critical toward the arguments of human beings we start insisting that everything -- including the irreducible Savior -- must be reduced that way. We start glorifying our own capacities as a species for reason, and we forget that what we have managed to reason in the brevity of our existence on one single planet is a trivial fraction of the universe (or universes) out there. We pride ourselves on our deductive ability when we cannot get anywhere interesting regarding the cosmic questions of life, the universe, and everything, without use our fallible powers of induction to choose our assumptions and what we will not contest today even when we dig down to "first principles". We forget that a first principle is never provable by deduction. A first principle is merely what we admit that we can't contest as truth/non-truth.
In short, man cannot reason without a stance of faith. You want to call your stances of faith better as you interpret the Old Testament and New Testament's relative importance in determining our salvation, fine. You want to emphasize Peter more than Paul, or John more than Jeremiah, I'm not going to contest that because I know my understanding of His meaning is going to be incomplete and you may well have better insight into the incomplete bits than I do.
But if you (or anyone) tell me that God isn't real, or if you tell me that Jesus was just a carpenter with a high moral code, then I'm going to dispute it to my dying day. Because that, my friend, is reducing God and Jesus to a religion, to something that is only real if the human mind can be encompassed by it. There may be or have been a million people in the world, ten million, a hundred billion, of people who are better at the history of the Biblical world than I am. Heck, maybe people like Ayn Rand, Christopher Hitchens, and Richard Dawkins, and all the other great atheists and believers in "reason" are all among them.
And that wouldn't change a thing. Because you could line up all of the greatest practitioners of reason in human history, from Galileo to Ricardo to Huxley to Popper to Einstein to Hawking, and all the results of their reason together, and you still can't debunk God without taking some stance of faith in those giant's sampling ability and in the uncertainties of sampling.
That's why I will celebrate Christ's birth as I've tried to do every year since I became more serious about my faith. Not because I know when the right day is. But because it's important that I celebrate that birth as having happened as the third most important anniversary in human history.
(There I go lecturing again. Argh. But whatever else the above lecture is, I believe it not a lecture on religion. I believe it a lecture on faith.)
And do not be conformed to this world, but be transformed by the renewing of your mind, that you may prove what is that good and acceptable and perfect will of God.
Romans 12:2 (NKJV)