How do I deal with the belief that maybe... Just maybe... God exists and I'm...

[Jehanne]

Jesus didn't make that distinction so this is just pointless pettifoggery.

The Greek word for "belief" there had connotations of trust, reliability, and being persuaded by argument. 

https://en.wikipedia.org/wiki/Pistis

So it normally looks to me as if "believe in Jesus" means to trust him or to be convinced that what he's saying is true. 

Do you know of any texts in which this reading is untenable? For example, a text in which the only reasonable interpretation of πιστεύεις is belief in the existence of something?

For the umpteenth time, the words of Jesus do not exist, as he did not write anything. Socrates did not write anything either, but at least scholars have Socrates words vis-à-vis Plato, although, it is debated how much of Plato's writings are Plato and how much are Socrates. None of that exists for Jesus of Nazareth.