But if you forget for a moment your semantic argument, you find that is actually very hard to forgive somebody that doesn't admit to have done something wrong.

Yup — that basically requires extremely high levels of compassion and unconditional love, something we can all aspire to, but in reality and in general forgiveness tends to be conditional.

That is the entire point of the Christian practice of forgiveness. It is unconditional.

It is a practice, something you do in spite of it being challenging.

The practice itself is mostly for the forgiver, not the forgiven. Try it and watch your heart soften and open.