It does make sense. Those three sources of evidence can have some overlap. We cannot for example retest that an ancient battle took place even though we know it did. Not everything is testable over and over again. Same applies here.
> But there is more than one version of Islam, isn't there?
What do you mean by version? Sunni vs Shia I presume? Yes, but we apply the same standards to arrive at truth. Furthermore, all "versions" have the same core: God is one, we all have the same Quran, we all have to pray, fast, etc. We all submit to God. Any other differences we can discuss to arrive at the truth.
> It also means you believe there cannot have been any mistakes in any of the writings.
We have an Isnad system in Islam, something that does not exist anywhere else in either faith systems or even secular systems. It gives us absolute confidence in being able to know if texts are valid or not. If we apply those standards to the Bible for example, it would immediately be rendered as weak (we don't know who the bible authors are).
The Quran is 100% preserved. The authentic Hadiths are 100% preserved. It depends on what you mean by "stories". This is a deep topic that I would not be able to do justice over a post, you'll have to watch an introductory lecture that talks about these things. Furthermore, there is a very specific process in how we go about deducing rules from Quran and Hadith. These are the major schools of thoughts (Hanafi, Shaf'ii, Maliki, and Hanbali).
> If there is no room for errors then there is no room for improvement.
Not true. There are many nascent topics that we have been able to derive rules for based on the approaches I mentioned.
> You don't accept the possibility of errors.
Not true. We know and classified Hadiths according to their authenticity, we have Hadiths that are Mutawater for example, all the way to falsely attributed Hadiths which are fabricated. We've done the work a long long time ago.
> There have been a great number of religions throughout thousands of years. Or iterations and versions of the certain religions. I find it arrogant and shortsighted to believe that your version is the final one and no other that comes after it will be better.
The Islamic view is as follows: God sent prophets and messengers to people throughout history. We accept Moses, Jesus, Noah, Abraham, Lot, Saleh, etc. Prayers of God be upon them all as prophets and messengers. There were many whom we don't have the names of, we're talking hundreds of thousands if I recall correctly. They all came with the same core message of the Oneness of the Creator, and to worship him and submit to him. As such, the followers of those prophets and messengers were called Submitters (Muslims in Arabic). Over time, those messages got corrupted. For example, Jesus Prayers of God be upon him, never claimed to be God or son of God, however, over time some people attributed this false idea to him, and we have the present day incarnation of Christianity.
Islam is the final revelation from God, and is preserved. It supersedes everything before it. Muhammad Prayers of God be upon him was sent to all of mankind until the end of time. He brought with him many miracles, and foretold many events in the future, some of which are happening today before our very eyes.
We have to accept all of those prophets that God sent, otherwise we're not Muslim.
On the other side, there are definitely man made religions (e.g. satanism) and many self proclaimed prophets that were not sent from God. Obviously, we don't give those people any credence, and we were warned about them in our texts.
Also, I've heard the fallacious argument before. Just because there are many religions does not mean that one of them isn't true. This is the genetic fallacy. Something that Dawkins and his ilk do all the time without realizing it.