Definition of faith

I have often heard you claim that a person who feels that God exists is an atheist with feelings, because he is not making a factual claim about reality. But can it be said that a person who does feel (and does not understand with his mind/intuition) that God exists but thinks that his feeling does reflect the ontological truth – is defined as a believer? Ostensibly, this is just like a person who comes to the conclusion that God exists through arguments but has a logical bug in his argument or one of his assumptions is unreasonable. He is mistaken in his judgment, but given that this is his conclusion – he can be defined as a believer.