Facts:
I won't offer a dictionary definition, since Apognophos is talking about how people can have their own definition of the word.
I will say that the factual truth of a belief (or a statement) does not change with the number of people who believe it. Santa Claus is not more factually true for pre-schoolers than for teens. Generally, the theory of evolution is not more true as more people believe it, nor would it be less true if less people believed it. It is either true or false.
Some people start picking it apart and claim to (or actually do) find a false statement within and then attempt to throw the whole theory out based on that small "fact" being wrong.
Tying back to the title of this thread, if someone states that something unproveable is a fact in order to feel that they have won some argument, then it just goes to show that "Facts" don't mean much if a person refuses to accept them.
You can be peaceful or minimize name-calling with people that refuse to face the facts. It isn't really approaching an olive branch in sincerity when you want every last vestige of it's symbolic meaning to spell out that it means you can state ridiculous things as if they are facts and the one who offered the olive branch won't even question what you said.
For some people, the facts of their life outweight the facts of science. For thousands of years, there was no science; only personal experience. Man got by in the world according to that experience. So to expect everyone living today to re-define "fact" to mean something abstract like 'objectively verifiable tenets, propositions or statements that are internally inconsistent' is not reasonable in light of the fact that we're all just hairless apes trying to make sense of the world around us.
If the above is not just poking fun, as I expect it is, the 'life-experience' definition of 'fact' is really the abstract one while the dictionary definition is a solid one.
Let's say that life experience taught our ancestors that a tiger is hiding in the bushes, instead of understanding that the wind can sometimes move the bushes. Those ancestors survived longer than the ones who refused to believe the idea that a tiger was factually hiding in the bushes and only believed that this was only one possibility. The second group did pretty good but were occasionally eaten by tigers. More of the first group got to spread their DNA.
So, genetically, we inherited the false belief that a tiger was always hiding in the bushes instead of the factual belief that it was a rare event. Just because it was wrong doesn't mean we didn't benefit from our beliefs. But eventually, man had to stop being afraid to walk in the jungle and had to overcome his inherited ideas about facts. Otherwise, we would have stayed just those hairless apes.