I think you are oversimplifying things and making massive assumptions.
You are aware that jihadists are killing MANY MANY more actual Muslims than westerners? How does this gel with your logic? Especially the one about the Muslims lying to non-believers. Bearing in mind also that the Jihadists doing the killing are in the minority in the places its happening.
Individual factions at war and constant power struggles don't negate that the entire religion in and of itself is barbaric at its very core.
In fact, it only further proves it.