I don't think it's that complicated.

It's simply a way of saying, "The God I believe in". It acknowledges that not everyone has the same views. (Depending on tone of voice, it's either very respectful or pretty snotty.) The same is true for expressing one's personal truths. It's just a way of saying, "This is what I think" or "This is what I believe". It's just oral shorthand. Instead of babbling on and on with, "My personal viewpoint, and you may disagree, which is okay, is that ____________".

"My God" or "My truth" assumes a maturity sufficient to understand the nuances of language and communication.