Attention agnostics and atheists: what constitutes a belief (see details)?

Agnostics and atheists sometimes challenge each other, but I think that oftentimes, many of these people’s viewpoints are the same in all ways except their definition of belief.

In terms of religion, I think there are two general definitions of belief that people use, without realizing that others don’t see it the same.

The loose definition of belief. If you think that one explanation is more likely than the alternative, that constitutes a belief. Believing is nothing more than recognizing probabilities. To draw an analogy, if you toss a penny twice, you might say “I believe I won’t get tails twice in a row” since the chance of getting tails twice in a row is only ¼, so it’s not very likely to happen.

The tight definition of belief. You recognize that one explanation may seem more probable than another, but that the less probable explanation still has some value of probability, and therefore may still be true. You don’t feel confident enough in any explanation to so far as to throw your belief behind it. Beckoning back to the penny analogy, “tight believers” wouldn’t say that they believe that the penny won’t land on tails twice in a row, because they know that it may well do just that, even though it’s more likely it won’t.

Of course, this is all complicated by the fact that all probabilities involved are not known numbers, but conjectures made by humans given their interpretation of the available information.

Which definition do you follow? Or do you fall somewhere in between? Do you think these different ideas of what belief is could be the only thing differing between some atheists and some agnostics?
More thoughts to come from me in the comments. This is long enough. XD