It really depends on the way the Inquisitor views the world and how much of a political animal they are. Take the two different approaches of my Dalish Inquisitors.
The first believed whole heartedly in his elven gods but had to admit they hadn't done much for his people in the last two thousand years. He admired Andraste, just as he admired Shartan, so wasn't totally against being called her Herald as he didn't regard that as a making him holy. Nevertheless, if anyone had asked him, which on the whole they didn't, he would have just as easily believed it was Mythal who was behind him in the Fade and returning to her People as Andraste (since he has Mythal's vallaslin on his face). Much of the time when asked about what he believed, he would answer "no comment". Eventually he just sighed and stopped fighting it, since everyone seemed happy to believe what they would. When they discovered in the Fade that it wasn't Andraste, he wasn't bothered at all and virtually said "told you so". By the end, he isn't sure what to believe any more. His estimation of the elven gods has gone down and so he takes his lead from his lover Dorian, it is reassuring to think that there is a Maker out there somewhere because otherwise you may as well believe in nothing. After JoH he has renewed disgust for the Chantry.
The second was already cynical about the elven gods when he left his clan (probably the reason the Keeper sent him). He decided that it might be expedient to go along with the general feeling, although I wouldn't call him a believer (more an agnostic believer). From the game mechanics point of view, though, when asked to declare either belief or non belief after Haven, I went with belief to see how different it would be. He also used the Andrastrian throne, because he acknowledged the value of symbols, which got approval in some quarters. When asked if he had a message for the faithful he declared that he spoke for the Maker because having done a few controversial things, it helps if the people think you have the Maker's backing. When it was revealed that it wasn't Andraste in the Fade, he suggested to Mother Giselle that perhaps they ought to tell people the truth, but she was adamant they shouldn't. So he just carried on, showing people mercy in the Maker's name and waiting for the day when someone calls him out (or the Maker hits him with a thunderbolt). At the end he was able to declare "I'm the Maker's chosen" to Corypheus in front of everyone just before he wiped the floor with him, so at the moment he's riding high on his reputation. I await to see whether things pan out differently in JoH for a believer.
So, two Dalish Inquisitors, two different approaches but both use the title to further their own agenda; it is just the first is more cautious in how much he uses his reputation than the second. Both fully expect the people to turn on them eventually because of historical precedent.