I have no arguments about your Arabic translation. Languages have nuances and it’s often necessary to give context, rather than simply translate word for word. You’ve given context, and I suspect your translation is probably more accurate for the connotation.
But, you other point. Nothing is more indicative of religion than the place you were born. You were born into a Muslim family in a largely Muslim nation, so you have been raised to be Muslim with a faith in Islam. Most Hindus are Indian, Most Europeans are Christians, etc.. Each thanking their own god(s) for being born the way they were and believing they have access to the ultimate truth.