Learn English – n adverb in English, meaning “with an interest”

I want to make a distinction not well covered by the other answers. In your example definitions, we find two different (very different) sorts of verbs: v.tr and v.intr vs. aux.v. It's the same word shape, but the meanings are almost entirely separate (you can sort of see how the one led to the other, but this is not relevant aside from historical interest).

aux.v is the auxiliary, or modal will, when it is used to as part of a larger verb form, such as will be, or will have been built. This is not the word in question. Exclude all of these from your choices.

We are thus left with:

v.tr.
1. To decide on; choose.
2. To yearn for; desire: "She makes you will your own destruction" (George Bernard Shaw).
3. To decree, dictate, or order.
4. To resolve with a forceful will; determine.
5. To induce or try to induce by sheer force of will: We willed the sun to come out.
6. To grant in a legal will; bequeath.

v.intr.
1. To exercise the will.
2. To make a choice; choose.

or more simply:

tr. & intr.v.
To wish; desire: Do what you will. Sit here if you will.

Any of these may be applicable, depending on context, though some are more common than others (generally, the higher ranked are more common).

A fact does, in fact, have to be the truth at the time you're using the word.

By 'truth', I mean something you believe to be true (due to any of several possible reasons).¹

Consider: "The number of planets in the solar system is eight." A few years ago, this was not a fact. It is now. (Just an example, don't attack the example.)

[EDIT: Before 1917, people thought it to be a fact that the atom was the smallest particle of matter. Today, it is a fact that it isn't, and we were wrong in thinking it was... I'm including this example to show that in light of the new fact, what we thought was a fact previously, can be rendered untrue for that time too; something @Jay pointed out my previous example didn't specify]

When a jury convicts a man, it's a fact that he's guilty. If he's later acquitted, it's a fact that he is 'not guilty'. (As far as the public is concerned. Individuals who actually saw the crime might know, for a fact, whether it's true or not)

What I'm trying to say is that the word fact is used for what you know (or sincerely believe) to be the truth or what is widely believed to be the truth at the time of speaking. Facts are subject to change.

Something is not a fact if you know/believe it to be untrue or if it can be easily be shown to not be widely believed at the time.

The two can contradict. That's when myth comes into play.

Tom: "Interesting fact: you can see the Great Wall of China from space." (Widely believed).

Neil: "As a matter of fact, you can't. That's a myth." (I went to space. I know better.)

¹ _{Please note that I'm talking exclusively about the cases where you use the word fact. In those cases, I infer you strongly believe it to be true.}