A "bygone" is a thing that happened in the past. A definition given of this idiomatic expression is:
To ignore or disregard a past offense (when dealing with another individual
Thus, you've already forgiven that past offense which that person, and presumably you want to get back to being friends.
As far as I know, this phrase is always used in reconciliation. For example, when two friends fight(as in your case), and they come back together, their conversation might run something like this:
A: I'm really sorry for what happened last week
B: Come, come, let bygones be bygones. What's done cannot be undone. Let's be friends again.
A: Sure mate!
In my personal experience, "let bygones be bygones" is used for reconciliation(your first interpretation)