I think you lost me when you assumed "faith" ought to be a meaningful word with a coherent definition.

I think the best definition to give for faith is a practical one: faith is the word people use as a combination semantic stop-sign and applause light when asked why they believe in religion. If someone then goes all philosophical on them and asks them what exactly they mean, they then use whatever plausible explanation seems appropriate.

I've heard faith described as:

• exactly the same as inductive reasoning; thus, you have "faith" that the sun will rise tomorrow.
• similar to the concept of credibility; the Bible's always been right before, so I imagine it will be right on this one issue here, even though I'm not exactly sure how.
• similar to the concept of trust: God is a good being, I'm sure He knows what He's doing.
• a direct meddling of the Holy Spirit in the internal workings of your brain, so that you are able to mystically come to the right answer about whether or not to believe in religion despite insufficient evidence.
• a decision to follow whatever idea seems intuitively true or most pleasant, and to give evidence less weight than these intuitions
• a decision to believe something even though you have no reason for doing so; also the thing those people who ask the obvious followup question "but why did you choose to believe that for no reason, as opposed to something else for no reason" should feel bad for not having.
• Not comprehensible to the limited human mind so don't talk about it

I don't think any of these definitions are "the correct definition"; I just think they're different ways that people in different situations and with different degrees of philosophicalness cash out the idea of "I believe in religion and you can't tell me not to and I feel pretty good about it"

As such, I don't believe there's a concept called "faith" which it is necessary to distinguish from theory in the first place.