What is weapon damage exactly?

I am a fury warrior, and I use Ramaladni's Blade of Culling and Justicebringer. Ramaladni's Blade of Culling has more damage than Justicebringer. But when i equip them, I see bigger damage in the Melee window when Justicebringer is in my main-hand. The question is, what is weapon damage? My main-hand weapon's damage, or the damage which I see in the Melee window? I must know this, because my spells (like Slam) use my weapon damage.

So weapon damage is actually the result of the weapon damage + your attack power once equiped (cause the weapon most of the times has attributes like Strength or Attack power all together, that add up to the resulting weapon damage).
Now...I'm not 100% sure of this , but i am to like 90% just for the sake of humility, the damage pointed out by your skills when they say "weapon damage" is the one that you see on your "melee" tab under character tab. not the damage of the weapon itself.

It's has to do with weapon normalization. Auto attacks are not normalized and benefit from slower speed weapons. Rend, deep wounds and heroic strike also benefit from slower speed. Instant attacks are normalized to 3.3 no matter what the weapon speed is.

Use justicebringer main hand because it will give your more damage with heroic strike and deep wounds. Instant attacks that are normalized to 3.3 are based on the weapons dps. Your slams will hit slightly harder but it won't outweigh the loss on hs and deep wounds.