What I see is how you use facts and beliefs in the wrong way.
This is a belief:
something believed; an opinion or conviction:
a belief that the earth is flat.
confidence in the truth or existence of something not immediately susceptible to rigorous proof:
a statement unworthy of belief.
confidence; faith; trust:
a child's belief in his parents.
a religious tenet or tenets; religious creed or faith:
the Christian belief.
This is a fact:
something that actually exists; reality; truth:
Your fears have no basis in fact.
something known to exist or to have happened:
Space travel is now a fact.
So, since I have seen that the sun comes up every day of my life, I think (not believe, for my thinking is based on prior facts) the sun will come up tomorrow. If it doesn't, then I will change my thinking. Since my employee has not been late for an entire year, I think she will not be late tomorrow; I don't believe she will, since I have facts to back up my thinking.
I believe the Yankees will win the world series next year. I have facts that suggest they win the world series more than any other team. But I don't 'think' they will win this year for there are no facts yet that they have the best team or the luckiest team, so I believe they will because I chose to.
A belief does not have to have any facts whatsoever to be believed. A fact has to have some basis in truth and it is demonstrable as being a fact. I don't believe 2+ 2=4, I know it equals 4, for I can demonstrate that that is a fact. Some facts may be involved in a belief, but that is not necessary. I can believe the earth is flat--don't need any facts to back that up--and I can then have faith in my beliefs. A theist can believe jesus of the bible existed, and they can be totally sincere and have the most faith of any human ever--but there is not one fact that would support that belief and faith. But that does not stop the theist from having those beliefs and faith; facts not needed.
Even the dictionary definition of belief contradicts your re-definition...
noun: belief; plural noun: beliefs
an acceptance that a statement is true or that something exists.
"his belief in the value of hard work"
something one accepts as true or real; a firmly held opinion or conviction.
"contrary to popular belief, Aramaic is a living language"
synonyms: opinion, view, conviction, judgment, thinking, way of thinking, idea, impression, theory, conclusion, notion
"it's my belief that age is irrelevant"
a religious conviction.
synonyms: ideology, principle, ethic, tenet, canon; More
doctrine, teaching, dogma, article of faith, creed, credo
trust, faith, or confidence in someone or something.
"a belief in democratic politics"
Its true someone can believe in something for no rational reason minus any evidence such a position is true. That doesn't dictate how belief works or that it always is blind belief. So, since I have seen that the sun comes up every day of my life, I think (not believe, for my thinking is based on prior facts) the sun will come up tomorrow. If it doesn't, then I will change my thinking. Since my employee has not been late for an entire year, I think she will not be late tomorrow; I don't believe she will, since I have facts to back up my thinking.
You make my case for me. You think the sun will rise because you believe it will rise. Prior to it rising you don't have a fact it will rise only a belief. Your belief becomes fact once the sun rises.