It’s a common definition amongst New Atheists that faith is merely belief without evidence. The occasional theist will accept this, often going further and purporting this to be some sort of virtue, but many Christians will reject this. They will argue that faith bears some relation to evidence, reason, logic, or even all three. But does that make sense? I don’t think so.
Let’s take the most populous religion in our culture: Christianity. If faith was more than belief without evidence, we should expect to find people with Christian beliefs that originated from somewhere other than the bible. That is, if faith is just a synonym for reason or logic or evidence, then a person ought to be able to discover all the information necessary to finding Christ.
Think about it. Calculus has been discovered at least 3 times (twice in Europe and once in Japan). The history of chemistry tells us certain elements have been found by several different people (usually with just one getting the credit). Atomic weapons have been created by multiple nations. All these things happen independently of each other. And why? Because math and science have methodologies behind them that progress on the basis of logic, reason, and evidence. Discoveries can repeat themselves in math and science. This is what we should expect of a type of inquiry that is more than belief without evidence.
When has faith produced the same result in independent people at independent times? Has anyone come to accept Christ in their hearts without the bible? Has anyone even come to know anything of Christ without the bible? Why is it that we don’t have any recorded instances of a Chinese person in, say, 80 A.D. writing about the Christian Savior? The answer is simple and obvious. It isn’t possible to discover anything offered on faith except through faith. To even know the name Jesus Christ, the original source of that name and of that man is always the bible; the original source is never found freely in the world, completely independent of the known Christian traditions. A tribesman in a remote part of the Amazon jungle will never know anything of what it means to be a Christian, no matter how hard he searches, less he find himself a victim of missionaries. Beliefs found on faith are beliefs without an evidential basis. Indeed, faith is nothing more than belief without evidence.
For the Sake of Science 
We all have faith to a certain extent. You can faith in a friend, and that faith is usually based on evidence. Theists would argue that their faith is based on evidence as well. As you point out with your scientific mind, that is not the case. But they believe it is the case. I’ve come to realize that a scientists view of what theists call faith is very different from how they see it. Which means that the disagreement isn’t so much about faith, but what constitutes evidence. For a theist they are quite content to say that anything in their holy book counts as evidence. A person like you and me would not. I’ve seen the Christian pseudo-science websites actually cite scripture as though it were an actual reference. There are tons of people out there who think all opinions are equal. A scientist would never say that. So there clearly is a misunderstanding as to what constitutes real evidence. There is a misunderstanding as to what the difference is between an opinion and a conclusion based on real evidence. There are people who think anecdotal evidence is as equally valid as statistical sampling. The law system still treats eye witness evidence as more important than DNA evidence. So when theists say they base their faith on evidence, I believe that they think that. I just don’t think they know what evidence actually is!
Having seen this discussion before in different places, I have seen it mentioned that confidence would be a better word for us to use. Faith can be turned around and used to proclaim that anti theists are indeed faithful to some degree, even when we know damn good and well that is not the case.
Evidence, is a word that needs to be pounded into theists heads. Evidence is not a warm fuzzy feeling, a belief, a liitle black book, or what the preacher or other authoritarian figure may have said. Evidence means much more than that, and there really isn’t much use trying to discuss anything with theists until they understand that distinction. Evidence is to me, a word that encompasses a multitude of facts that when put together can be understood by any rational thinking person as a foundation for a truthful statement or understanding. The more facts you have, and the more rational minds that agree upon those facts, the better the evidence.
Now lets describe rational vs. delusional.