So according to your theory, people God has chosen to be regenerated will believe and be saved, and those God has chosen not to be regenerated will not believe and will be condemned.
Correct. And it should be noted that this is not simply my private theory or interpretation. This is the interpretation of the Reformed Tradition.
Therefore, God has not really given people the choice to choose whether they can be saved or not, is that correct?
I would never put it like that.
In case you're not familiar with philosophy, we're operating from two different notions of freedom. You seem to be promoting
libertarian freedom whereas my understanding would be
compatibalist freedom. We don't need to get into that, but it's useful background info.
I would say that people make free decisions in the sense that they are not coerced or restrained. A sinner sins freely. No one twists their arm into sinning.
Think about it like this. A person in prison is not free. They desire to go home and be with their family but they can't. They are restrained. Likewise, a person with a gun to their head being forced to perform an action against their will is not free. They are acting, but their decision is coerced.
When sinners sin, no one has a gun to their head. Likewise they are unrestrained (to an extent). God has given them to ability to do as they please. So they freely reject him.
But when God regenerates a person he gives them new desires. Suddenly they desire to obey God. They then become free to leave their life of sin and to obey God in a non-coerced and unrestrained way.
This is why Paul says in Romans 6 that once we were slaves to sin, but now we have become obedient to God from the heart and slaves to righteousness.