Proof exists in the reality experienced by those who experience the proof, the rest is conjecture.
It's the experience that makes the reality for the one experiencing.
"No atheists in the foxholes."
So what if somebody that is atheist calls out to something at their weakest point? Panic can make people do crazy things, believing in god being one of them. There very may well be something out there, but the chance of it being any specific god are so slim it's not even worth trying to pick one. On the edge of death you really have nothing to lose so you might as well hedge your bet, doesn't mean you actually believe in it once you're thinking clearly again
Well, I'm glad you are making a start.
Someone who is in a panic, and calls out to God, isn't calling out to any specific God, necessarily. They are calling out to THE God.
THE God resides in all of us to some extent. Few if any of us can interpret the inner workings of our own hearts and minds. Because of this, when we look at God in ourselves, our view isn't going to be clear.
The various religions might have their favorite god. Is the "thing" that they have, god or God? Nobody knows. It depends on the various person and his "view" of god/God. The person might not even know. But God does.
The best thing we can do is to examine the various religions, and hope that we will get enough of a response from the true God during our examinations, so that we can recognize Him when we find Him. Even the religion that has the best understanding of God isn't going to be completely accurate in what God is really like. This is reflected in the various methods of worship. The thing that we are looking for is to have a correct enough view of God so that He will judge us as believing in Him rather than believing in an idol or false god.