Great you seem to understand
It is funny how The Quran contradicts with what they (The Terrorists) say.
Btw, ''Jihad''.. Many people believe it's like ''Keel teh infiduls!!11''
But no, There are many kinds of Jihad, Like the best Jihad for women is to do the Hajj (Go to Kaba, ''The building which looks like a black box).
Jihad can also be if you Smoke and you stop smoking, Then that is Jihad (If you try your best).
And in war it is only allowed if they attack you first.
Hope I explained it well to you xP