I chose it because I have some sort of ethereal ability to sneak up on people. Interestingly, I don't think I always had this ability, but possibly got it from a movie. I can't remember what the movie was✝, but in it a cop was describing a tense showdown with a person that had a gun or something. He described pulling back his presence and trying not to exacerbate the situation. When he finally spoke, he startled the person who had somehow forgotten he was there. I thought it was a pretty cool scene at the time, and after that, as if I had somehow learned the technique, I mysteriously started surprising people. Usually people will freak out because they never hear me coming. I can't really turn it off, and I have no conscious control over it, so it's not that useful for situations where I want to be stealthy. Which brings us to the next part.
There is another aspect to the appearing, and so there is another part to my presence. That thing about coming in like a god. I am also very visible. Especially in classroom settings. Whether they liked me or (more rarely, thank goodness) hated me, teachers always took notice of me. People form strong impressions of me. I can't call it charisma, because it's not necessary positive. The problem is only worse in Japan, where I am very visible (yet I still sneak up on everybody here). Anyways, that's how I appropriate the god portion of the idiom. Then I disappear...
In daily life, I use my odd foreign part time teacher status to appear around the school, investigating how things work or being seen reading books. I also don't like attention that much and so do my exiting like a phantom quite often. I remember someone in college complaining that I thought I was Batman. Sorry commissioner, no time for small talk.
*: Some other definitions--incidentally this is one of the first four character idioms I ever learned in Japan:
✝: If anybody know that movie, let me know.