I think we can take 'faith' in a literal definition, where one has faith, or trusts, in a decision, as you have pointed out. In the context of religion, we can have 'faith' in our decision, or trust we made the right decision, of choosing whatever entity we do or not deem sovereign in our existence.
However, this doesn't necessarily require the need of proof, as you mention.
Yes, God provides incentive to believe, to trust, to have faith in him in the form of these relatively great feats, but that is because the target audience has been proven to be hesitant of having blind faith, particularly since this is the deity that left them to be enslaved by the Egyptians and made to wander around a desert for a generation.
I, myself, am not sure if this is a proper way to approach a religion, but it may be important to point out the existence of 'blind faith,' where people don't feel the need to directly witness particular feat to feel compelled to believe. This may be from, but not limited to, the use of anecdotal evidence when being informed of said religion or being immersed in a community that is devoted to a particular religion to 'convert' an individual to a particular religion or sect.
