Re: "I'm right and you're wrong!" ... "No. You're wrong and I'm right!"
The only way to know who has arrived at the truth is to take a look at the concept of God. The concept of God in Islam is complete, whole, and it is clear. This is the most significant reason why Islam is the only religion which I can believe in. All other religions fail to properly explain Who God is. What we know about Allah is that He has no short-comings, but the presentation given about God in other religions falls short of a Perfect Being.
As for scriptural interpretation, it is agreed by scholars in Islam that the Companions (ra) of the Holy Prophet (saw), while learning the Qur'an, learned the Tafsir for it as well. They did not progress in memorizing any more Qur'an, until they knew the Tafsir of the verses they already memorized. The Holy Prophet (saw) himself taught the Tafsir of the Qur'an. There are definitely multiple meanings to the verses, but he (saw) taught the Muslims these meanings. He was the one who understood the interpretation of it best. Also, Hazrat Ayesha (ra) said in the Ahadith that the Holy Prophet's (saw) character was the Holy Qur'an. This is the most logical conclusion whether she even stated that or not, because the Holy Prophet (saw) is the best example for the Muslims (33:21).
Therefore, there is a firm basis for understanding the Qur'an in Islam. --- So Islam has a safeguard for its problems.
As for Christianity, I chatted about Christians on the interpretations of the Bible, and none of them know of any authority which is looked up to for the basic interpretation of its verses. If I were Christian, this would have been discomforting. It is better to have a rigid basis for interpretation, rather than an open-ended Book which in some cases has literal applications, and in other cases parables, but with no way to know what the actual meaning is. In fact, this is why Atheists keep bashing Christianity in the current times we live in. I don't agree with their poking fun at it, but since there is no solid basis for how to interpret the Book, there is really nothing Christians can do to ward off their insults. So open-ended interpretations, in my view, has caused a lot of problems for Christianity.
This truly is not the case with the Qur'an. The interpretation of the Book is based first and foremost on mutually supporting verses, which explain the meaning for each other, and after that there is the Traditions of our Prophet Muhammad (saw), what he said, what he did, and what the Companions (ra) did after him. To understand the ancient Arabic expressions mentioned in the Holy Qur'an, there is the classical Tafsirs, the dictionaries of Arabic which preserved such explanations and the old Arabic poetry from pre-Islamic times - that highlights different expressions and how they were used in the past. Besides this, there have been countless scholars throughout the ages who preserved these meanings and transmitted them, explaining them for all kinds of people to understand, due to the spreading of Islam to different cultural backgrounds. Thus, the pious people and scholars of Islam did their best to preserve the meanings of the Qur'an, and did their best to spread these to everyone, in speeches and in writing. Great care has been taken for the preservation of the meanings of the Qur'an, so much so that any Muslim who learns Arabic can learn them and find a teacher to help them learn these.
Again, I do not know of any such preservation methods for Christianity throughout the ages. Islam has been safeguarded by Allah, and He has never lifted His Hand from this since He revealed this religion to His Messenger (saw). Finally, the reason why Muslims are so convinced that Islam is the right religion, and legitimately, is that this religion is the final religion to mankind, with the final and complete law, furthermore, the Prophet of Islam (saw) is the final Prophet to mankind, of a religion which is specifically meant to be spread to all of mankind, thus, Islam is the true religion. It confirms the truth of all the past revealed religions by God, such as your religion, Christianity, Judaism, etc. It is the final religion for mankind, and because its law is final and abrogated the laws of Christianity and Judaism, therefore, it is the only religion for mankind. Christianity is correct, as far as what Jesus (as) actually taught in the past, not the version that exists now. However, if the Prophets of the past were alive today, or during the time the Holy Prophet (saw) was alive, they would have followed him (saw). They were Muslims in their times, and for whatever future Prophet was to come after them, all of the Prophets covenanted that they would follow that Prophet when he comes. That refers to the Holy Prophet Muhammad (saw), because his religion is the religion of all the past Prophets, he brought nothing new. However, his religion completes all of the past religions. Allah sent down His final Law, and thus it is for every man and woman to follow it. It has nullified all past religions, even though they were true.
. The Christian will say the source of his belief is the Bible and its related teachings.