I definitely change the name, too. Not as creative as Mesix was in that department, but I had fun smashing together some High Valyrian words to form a name that made it feel more mystical. It just wouldn't feel the same to hear someone asking or converting to Confucianism. It's odd, and it makes me wish the AI named their own too (very generically, of course).
And yes, I know what you mean about making up our own: that's the main reason why I enjoy religion in this game. I wish you could also create your own civilization besides just renaming it and the cities! I really do wish they had some more icons, too. Throw in some of the ones you mentioned but also some completely fictitious ones. Other than that, my main joy comes from attempting to create one world religion (Rome and Mecca are both under its sway, e.g.). Unfortunately, I'm bound to end up a hypocrite by having to cripple another's religion in order to do it.
I've been such a tree-hugging pacifist in this game, but the AI does a good job at pushing my buttons to the point that I'm about to start over and paint the map with holy warriors and petroleum. If anything, it'd be good for the experience, which I can use in this game.