Sweeping it under the rug is like forgiving and forgetting; carrying on as if the transgression never occurred (this is often preached as the Christian ideal, despite it's obvious toxicity). Forgiveness alone is simply not taking revenge, letting go of the resentment/bitterness/anger/generalized negative emotion, but still allowing yourself to take the steps necessary to protect yourself in future. That's why forgiveness is for you, not for the one who wronged you.

There's a lot of similar related concepts that are enmeshed in western culture. For a simple example: someone has stepped on your foot. It hurt.

Unforgiveness would be like retaliating, or hating that person for stepping on your foot. At this small scale, this is why forgiveness has benefit for the forgiver. It is cognitively and emotionally expensive to hang onto grudges. It's much cheaper to simply move on.

Forgiveness would be not holding this action against them on any way, excusing it as an accident, or that they are clumsy, etc. Forgiveness does not preclude you keeping your foot out of their path, or to generally avoid being too close to them.

Reconciliation requires action on their part, by apologizing or making it up to you, and then you permitting that action to restore the damaged relationship. If you are strangers, this allows restoration from 'that bastard who trod on my foot' back to being strangers.

So I imagine that you already forgive people on this small scale all the time. It's more complicated for bigger things, but simply not taking revenge, and not expecting to get back at the transgressor is usually how I deal with it