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There's no context independent meaning and depending on the context it's useful to use different notions.
I personally distinguish emotional beliefs, probabilistic beliefs and logical beliefs.
The native encoding in which beliefs are stored in minds is emotional and most of the beliefs people hold don't have explicit probabilities attached to them.
When it comes to reasoning about well-defined events in a structured way, probabilities are very helpful.
I'm programming I don't ask myself about the probability that a certain variable might be null. If I consider it a plausible state for the variable to be null, I rather handle it in my code regardless of the probability that I attach to that event.
That presented with a bet that essentially hinges on that thing being true, you take that bet.
Note that actions involve uncertainty and costs, so they involve a form of informal betting.
It largely depends on what work you want the words "belief" and "truth" to do.
We might say a belief is a proposition we reason from as if it were true (we'll get back to truth shortly, but I want to screen it off for now). In humans this means a belief is some kind of mental activity corresponding to some statement about the world that is integrated into the mind as part of the network of propositions used in reasoning. Beliefs can exist independent of truth values, though, as anyone who has even mistaken believed one thing and then discovered the world was otherwise knows.
We sometimes use "true belief" to refer to a set of propositions known to be true (known facts) to some entity, so then to believe a thing is true is to know it to be true. This runs into epistemological issues because it necessitates assessing whether something is true, and thus asks us to know the criterion of truth, which runs us headlong into the problem of the criterion. Although I think the problem of the criterion gets at the heart of what you are asking (which is ultimately grounded out in questions of existence), we can be pragmatic here and ignore it by taking a pre-formal leap-of-faith that we know how to asses what is true.
Then to believe a thing is true is simply to know it is how the world is.
To use it as an assumption when reasoning about the world. See Making Beliefs Pay Rent (in Anticipated Experiences).
Just listened to that about 10 minutes ago. Good sequence. When you say "assumption", I hear "a thing that is accepted to be true".
What changes in cognition when something is accepted as true?
In the Greek πιστεύω ( pisteuó ) is to believe, and is derived from πείθω ( peithó ) , which is to be persuaded of what is true. There are undoubtedly different strengths and types of persuasion, but I find that this understanding (that is to believe something is to be persuaded of the truth of it) is useful in all situations and contexts.