For the "believers": What evidence do you accept for the existence of God? How would you differentiate between possibly legitimate visions, miracles, etc., and hoaxes or misunderstandings?
This comes from a Torah-observant Jew. And before I answer, let me just say this: I don't need evidence. A soul has a natural connection to God as it is part of God. It is only when this connection gets clouded and plugged up that we need to "prove" the existence of something our souls know already. When people find something that proves a "story" in Tanakh or something, they have it backwards. That part of the Torah isn't true because they found some evidence; rather, that evidence had to be there because the Torah is true.
Three parts to the answer: what would I accept for evidence or what do I accept, differentiating between stuff, and your specific example.
Evidence I accept or would accept:
1. Prophecies in Tanakh. There are numerous prophecies all over Tanakh that have come to fruition, many of which are either discussing specific dates and people or things that seem to go against what would make logical sense. Nonetheless, they have come true.
I won't get into the details unless someone wants me to, but a few of them are the connection between Jews and the Land of Israel; the destruction of the First Holy Temple, the Exile in Bavel, and the construction and destruction of the Second Holy Temple; the miraculous fall of Sancheriv's army by an overnight plague; and the preservation and sanctification of the Jewish people and the prophecies regarding our population and punishments.
2. Extra-Biblical evidence and accounts of events in Tanakh. Recently, there was found a piece of writing from approximately the time of the Judges. The Hebrew writing not only points to widespread literacy in the Land of Israel in the 10th century BCE, but discusses the Judges which occurred during that time period. About a hundred years ago, several inscriptions in the Sinai desert were found discussing various events found in the Torah during the Exodus (kriath yam suf, the plague following the golden calf, alters for bringing offerings with names on them). The language was clearly Hebrew, but the script was very proto-Hebrew and from about the same time as the piece of writing just found last year--which would put it at around the same time the Exodus is supposed to have happened. Interestingly, the description of the parting of the Sea of Reeds does not quote Biblical passages and is written in a completely different style essentially from the point of view of a first-hand witness.
There have also been numerous artifacts found in Egypt that discuss Hebrew slaves not meeting their quotas and either attempting to escape or generally being unruly. There have been clay tablets with Biblical names from Genesis that date to that time period, as well as a very famous sculpture of a ram caught in a bush (from the story of the Binding of Isaac) that dates to the 20th century BCE--around the time it would have happened. This is not to mention the numerous things that have been found in the Land of Israel that confirm later events in Tanakh.
3. Modern events. If I were an atheist, the miracles of the Six-Day War and the Yom Kippur War would be enough to convince me of the existence of God, or at least that there is a Higher Power watching out for the Jews. Reading about a small nation being brought so close to utter annihilation and then experiencing miraculous victory after miraculous victory against multiple nations from all sides gives me chills.
Personal events also take a hand in this, as I have experienced a bunch of "hidden" miracles myself (when things just seem to go right when the odds are against it) and a couple of "open" ones (when something happens that seems to defy logic and/or nature in an obvious way). Additionally, there are daily personal experiences that are difficult to understand, such as feeling the Presence of God during prayer, having prayers responded to and answered in one way or another, etc.
4. Logic. While parts of our faith defy logic, the basic story and belief system is completely logical. There is a non-physical entity Who was the impetus for the creation of the universe. We could spend a whole thread just on the logic of believing in God.
Differentiating between possibly legitimate visions, miracles, etc., and hoaxes or misunderstandings:
1. Possibly legitimate visions. There is a difference between having a psychic vision and having a prophecy. A psychic vision is something that comes from a natural God-given talent. However, because they are derived from the human brain or senses, they are fallible and frequently at least partially--if not completely--wrong. Prophecy, on the other hand, comes directly from God and is never wrong.
There is no more prophecy. We have not had it for over 2,000 years. The book of Malachi ends with the following verses:
Malachi 3:23-24 said:
הִנֵּה אָנֹכִי שֹׁלֵחַ לָכֶם, אֵת אֵלִיָּה הַנָּבִיא--לִפְנֵי, בּוֹא יוֹם יְהוָה, הַגָּדוֹל, וְהַנּוֹרָא
Behold, I will send to you Eliyahu the Prophet--before the coming of the Day of the LORD, the Great and the Awesome.
וְהֵשִׁיב לֵב-אָבוֹת עַל-בָּנִים, וְלֵב בָּנִים עַל-אֲבוֹתָם--פֶּן-אָבוֹא, וְהִכֵּיתִי אֶת-הָאָרֶץ חֵרֶם
And he shall turn the hearts of fathers unto their children, and the hearts of children unto their fathers--lest I come and strike down the land to excommunication.
Anyone claiming to have prophecy at this time is first considered to be either lying or have a psychiatric disorder. That said, we do believe that there are bits of prophecy still given to people in times of need or in certain situations that warrant it; however, there are no visions from God that have national and future consequences.
2. Miracles. This one is easy. If something happens and is miraculous, it's a miracle. A miracle does not have to be something huge like the splitting of the Sea. The Book of Esther is all about how God performs miracles every day using the natural world He created. These hidden miracles happen all the time: when you just happen to be in the right place at the right time, or an unlikely serious of events occurs to your benefit, etc. That is just as much the Hand of God as 18 Egyptian soldiers being frozen in their boots and unable to fire on 2 helpless Jewish reservists during the 1967 war.
Of course, if someone is using the term miracle as hyperbole like we tend to do nowadays, that may not be such a miracle. But common sense is pretty good at discerning when someone is being dramatic or not.
3. Hoaxes or misunderstandings. I'm not quite sure what you're talking about here. Do you mean when someone does something and claims it to be the work of God even though he did it himself? When a Jew tells me a story, I just assume he's telling the truth. If I find out he was, God-forbid, lying to me, that doesn't shake my faith in God. I just wonder why he felt he had to lie about something stupid like that.
Your specific example:
It could be an open miracle, and it could not be. I wasn't there. The story could be exaggerated, the illness could have been non-life threatening and self-limiting, etc. Regardless, someone who was ill and got better experienced a miracle. God could easily just have let that person die.
Our problem is we are unable to see the miracle in just waking up each day. The fact that we go to sleep and God allows us to live another day is nothing short of a miracle. This spiritual separation from God is why we have so many problems in this world. There is a great video on G4 about how in the near future, everything is basically going to be in the format of some sort of game or points system. The speaker goes through a day in that kind of life, and at the end explains how being constantly recorded and tracked can inadvertently cause us to be better people for the sole reason that we don't want our offspring to be embarrassed when they look at what we did with our lives. The fact that we need all this technology to be good people just shows us how far away we are from God. He gave us a book about how to be a good person if we only follow His example, but we're so far from Him that we aren't even able to see it.