He is worshipped differently by each faction, High Elves worship him as a god of war. Dark Elves worship him as a god of murder. It's basicly the same difference between Khaine and Khorne. Khorne does not want murder, he wants blood shed in combat which happens in a war. So it really depends on whether Khaine prefers murder or war. He is probably depicted different by both factions.

As for the Khorne/Khaine thing, I'm pretty suspicious about that one. For one thing you don't pray to khorne for victory, you just fight and kill. Whether you die in combat, or win to keep killing it's not because you prayed to Khorne for victory, it is because you were the better warrior. Yes he grants gifts and magical weapons/armor, but you have to earn them to begin with. Chances are that 7 foot psycopath with the axe from the blood god that gives him the strength of a Bloodthirster did not pray for it. He most likely earned it by killing in combat.