Meaning of "let bygones be bygones"

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)

Let bygones be bygones is used in a conciliatory tone.So, yes, it means you're putting the disagreements behind you and getting back to being friends.

from The Phrase Finder:

So, there is a little more to the phrase 'let bygones be bygones' than to the more literal 'let sleeping dogs lie' or the old proverb 'let all things past, pass' that was recorded by John Heywood in his 1562 edition of Proverbs. 'Let bygones be bygones' uses both meanings of the word 'bygones' and means, in extended form, 'let the unpleasantness between us become a thing of the past'.

So I think, the meaning of the phrase is closer to your first meaning versus the second.

Forget the past, turn over a new leaf. The past is history!