The English language (and all natural languages) are sloppy, ambiguous and vague. But they still tend to work surprisingly well. It doesn't work very well when a specific, technical meaning needs to be discussed. That's why scientific, engineering and medical fields have all sorts of specialized words that mean something different from the way they're frequently used in everyday speech.
This also applies to philosophy. It's hard to have a deep philosophical discussion when you use a word to mean one thing and someone else uses the same word to mean something different. So, here's my small contribution. My take on what the word "belief" should mean in a philosophical context.
A belief is something that is held to be true.
Not something that is held to be true without evidence. That's faith. It's common in ordinary speech to use the two interchangeably. But I feel when precision matters, these definitions are the best readily available.
The biggest reason for that is that frequently, it doesn't matter why something is believed, only that it is believed. In such a situation, a more general definition is better. As far as I know, there's no better word for "something held to be true" than belief. Faith, trust, knowledge, etc. are similar but all more specific. Belief is good for it's generality.
Another reason is that it's not always so cut and dried if a belief is faith, trust, etc. What if the belief is held for a reason, but a bad one? What if there is evidence supporting the belief, but more evidence against it? Is it faith, is it trust? Well, either way, it's still a belief.