It's not totally clear how True Faith works, but it's definitely not just internal to the believers. The White God, whatever it is, clearly exists and exerts real power through its angels.
Picking an example at semi-random, Shiro didn't get his powers from being a Baptist. He got them because God (or perhaps his angels) decided to give them to him. I wouldn't be surprised if everyone with True Faith was in a similar boat. But for game purposes I'd totally allow a character to take Righteousness fuelled by their belief in the American way or whatever; whether or not it's what Butcher intended, it'll work fine in play.
If you ask me, the big mystery of True Faith is what exactly God is. The actual-factual miracles in the story make a number of real-life theological controversies a lot harder to answer.
I'm not really sure it's possible to discuss that topic without getting into touchy territory, though. Had a whole thing written out on the topic and deleted it because it was mostly applicable to real-world religion as well as Dresdenverse religion.
Here's a weird thought. Islam doesn't have a lot of symbols (it's a whole thing), but the first verses of the Qur'an that were revealed talk about how God taught humans with a pen. Would you let a sufficiently devout Muslim just pull out a pen and start warding off a vampire with it?
I'd allow it. It's not mechanically overpowered or religiously offensive, so why not?