-The actual war (with extended plot challengers, trophies, and prize shop) was the only 'war'.
-The second one was the first 'Battleground' event (this being the second), we don't know how long it'll go on for at the moment. What we do know is it's not a full site event/war, and appears to be a weekly event now, nothing special.
You won't receive a trophy for participating in these 'Battleground' events, but you will receive an avatar if you're in the winning faction (and win a min. 10 battles), a site theme, and a choice of a 'boon':
(you need an account to see links)
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There's pretty much a battle event for 3 days a week, with a choice of 3 different factions.
You must fight the challengers on the opposing factions' sides for it to count.
If you're on the winner's side, you'll get an avatar and a temporary 'boon' bonus until the next battle event.