I don't believe you are required to swear with your hand on the bible and I think that is the difference at the moment.
My understanding is that the practice is witness-specific. Christians swear on the Bible, Jews on the Torah, Muslims on the Koran, etc. IOW, your own Holy Book.
And, of course, the Constitution specifically permits a person to refuse to "swear" on anything, and to merely affirm their truthfulness.
I was just in court yesterday, and I was told to raise my right hand. He said
"do you swear that the testimony you are about to give is the truth, the whole truth, and nothing but the truth"
That's where it ended. I thought about answering "So help me God", but chose to play it safe and say "yes, your honor".