I think your view is being skewed by the religious use of faith - which does seem to be about beliefs. I agree that one can be faithful to one's beliefs (or principles). But if you think about common-or-garden phrases like " faithful friend", or "supporter/fan" or "husband/wife", or "servant" or "dog", I think you will see that in those cases, it is not about belief at all. It is about how someone behaves - different behaviour in each case, as required by the relationship in each case. "Faithful picture" or "account" are different, but obviously not about any beliefs. — Ludwig V
In my post, I already agreed upon the fact that the meaning of “faith” must be determined in the various contexts of their usage. Still, if the task is to identify some commonalities between some (not necessarily all) various usages, then one has to make some effort to abstract from a fine-grained analysis of each specific usage.
Besides, I also warned that my idea that “faith” is some sort of epistemic emotion was just a starting point in need of further elaboration, like the one you suggest:
In religious contexts, faith is also related to some normative practical engagement (which may include rituals and pious acts) by which we assess how virtuous and/or meaningful one life is. And also in ordinary usage, "faith" conveys some sort of informal engagement by which we assess people reliance, especially under test . So yes the behavioural dimension is also worth mentioning.
Still, what I would disagree with you on is the following claim:
if you think about common-or-garden phrases like " faithful friend", or "supporter/fan" or "husband/wife", or "servant" or "dog", I think you will see that in those cases, it is not about belief at all. It is about how someone behaves. Indeed, your putative counter-examples seem to be very much compatible with what I wrote. Beliefs do not need to be about what exists, their identity or properties, beliefs can also be about how people behave. For instance, when we talk about a faithful friend, parent, dog we are referring to the fact that these friend, parent, dog will act in ways we would expect (and approve of) from friends, parents, dogs based on passed behaviour. It is precisely because friends, parents, and dogs behaved in ways we approved of in the past, that we can believe they will do it again, and rely on it in our life (maybe even under daring circumstances).
"Faithful picture" or "account" refers to idea that certain representations won't betray expectations based on them, they can be trusted, I’m tempted to add "as men can be" (because those expressions can sound as a personification or a metonym).
By the way, I’m inclined to say that faith in an ordinary (non-religious) sense looks more synonymous of “trust” than faith in a religious sense (rhetorical nuances aside, i.e. “to have faith in” sounds more solemn or stronger than “to trust”), as if the religious understanding of faith is richer than that of trust.
But I think the religious use of faith is more complicated than it seems. In the Christian faith, the creed and signing up to it are very important. In other faiths, beliefs are less important. What matters most is behaviour - behaving according to the moral code, taking part in the liturgy and so on. Religion is only part about belief and only about belief as part of a whole way of life. — Ludwig V
Again I agree on that the religious notion of "faith" has a complex semantic and that involves behaviour. I also readily referred to it in my post with the expression
“normative practical engagement (which may include rituals and pious acts)”. But the idea that “beliefs are less important” in faith (at least, in other religions compared to Christianity) sounds rather a misleading objection to me. “Rituals” and “pious acts” concern people’s behaviour, what people say or do in certain circumstances. And behaviour, what people say or do, can be performed without having appropriate inspiring beliefs or, even, theologically elaborated or critically scrutinized beliefs, or even a satisfactory grasp of what it is believed (religious people can believe in mysterious things like the holy trinity, Jesus' dual nature, miracles, etc.). So yes, in some of these senses belief in religious faith can be said to be "less important" than behavior. The point however is that also in the religious contexts behaviour, especially in the long run or under daring circumstances, is typically taken as an indicator of the strength/authenticity of one’s religious beliefs. Indeed, if people would perform rituals and pious acts without believing at all in the creed that inspired them, maybe due to peer pressure or out of irreligious interests, I doubt we would take them as a the paradigmatic example of religious faith. Prophets, saints and martyrs… they are.
On the other side, the difference between Christianity and other religions you are pointing at may even lead us to not consider those other religions as religions if the element of faith in some supernatural/sacred world is remarkably lacking (e.g. Buddhism is considered by some more as a philosophy than a religion).