All the Calvinist that I am aware of define "belief" in the same way the Bible does - (and I have a great many of the brightest and most accepted Reformed theologians represented in my well read library).
They do not define belief that way. They say that men "believe" because God regenerated them and made them a new creation - one which was able to exhibit "belief" in the scriptural sense. That is quite a bit different than what you just said.
No Calvinist preaches to men that they must wait for some kind of regeneration in order that they believe.
The unregenerate are unable to understand the doctrine of regeneration even if it was explained to them much less execute some kind of waiting on God to obtain it.
There is "no one" who understands and no one who seeks God.
"There is none who understands. There is none who seeks for God" Romans 3:11
Salvation is freely and wholeheartedly offered to every man who hears the gospel. Every Calvinist believes that.
Many Calvinist do say (under the non- Calvinist inspired doctrine of limited atonement) that Christ did not die for the non-elect. Would that they would agree with what John Calvin taught about the extent of the atonement and not do that.
No - all men have a choice to accept or reject Him.
Any regeneration is because He extends grace to some who deserve only Hell and opens their ears and eyes to the truth they were not able to hear and see because of their own sin.
God knows who will accept or reject Him from eternity past. He regenerates those sinners He has chosen to show His mercy through in the ages to come. The other Hell deserving sinners He passes by and leaves them to their sinful unbelief.
Jesus gave you a parable that shows you exactly why this is fair - if you will but take His teaching to heart.
God gives some people what they bargained for (when they first sinned in this case). The just wages of sin is death.
To others He gives a better wage - eternal riches and glory in Christ Jesus.
No one has a legitimate beef about that. He's the owner of the vineyard. He makes the rules.
Some will, of course, pitch a beef to Him. And it won't even make it to the plate.
"Is it not lawful for me to do what I wish with what is my own? Or is your eye envious because I am generous?" Matthew 20:15
I wouldn't think so.
But if there is and they are offered forgiveness it would likely have to be because God became a cow and was sent to the packing house for their sins.
The doctrine of the Calvinist (which you are disputing here) is "unconditional election".
I would think you'd know that already since you are venturing to disagree with them.