I think if you asked someone right after they said OMG, "are you praising the name of God?" my guess is the majority of the people would give a sheepish "no" or they would be perplexed.
If, to use another's point, an athiest screams "OMG" in the middle of passion, you stopped them & asked "are you offering up a prayer to God &/or was that an acknowledgment of His existance?", they would probably reply that it's "just a saying" or you might get a sarcastic "sure". Of course if you stopped them in the middle of passion, they might have a few questions for you.
I think if you stopped every person that had the "God Is My Co-Pilot" bumper sticker what the intent of the sticker was, most would tell you they are making the statement that they couldn't live without God, thus proclaiming His greatness.
Here's one that you didn't mention that I've heard: "God's country". When people say this, I don't think they're saying that God's protection is solely reserved for a specific land. When people say this, I'm sure most people mean "this area is so beautiful & fertile, that
only the hand of God could have created such a place", thus again, proclaiming His greatness.
To get back to point at hand, there is a difference between: someone laying on their back, gazing at the skies & in amazement of the complexity & vastness of space utters "My God..." with a tone of reverance
& a woman showing off her new shoes & her best friend blurts out "OMG!" IMO the latter trivializes His name.
Let me put it another way, using the exact opposite effect: "bummer" It's a good word to sympathise with someone that lost their keys or stain a shirt. It would be horribly inadequate & inappropriate to use "bummer" to convey sympathy for the loss of a relative. It trivializes the moment.
Some people also say "OMG" that they would also say "Oh brother" complete with a

; let's say that Bob is a loved one (and for this hypothetical there is only one Bob in the world), you know he loves you deeply & there isn't anything he wouldn't do for you (provided it was good for you) & to you Bob is the greatest. Now let's say there's a group that has never met Bob or they don't have a high opinion of Bob & conciously or subconciously they use Bob's name for things, like "Oh Bob!" like they would say "Oh brother" or they say "Bob" the way Seinfeld says "Newman" & one day you overheard someone using Bob's way in a not-so-flattering way. Wouldn't you be a little offended & compelled to tell them how great Bob is? Wouldn't you be a little hurt everytime you heard Bob's name thrown around & you know it's not to talk about how cool Bob is?
Some Jews feel the name of God is so majestic & sovereign, they won't even use the name "God" unless they are speaking directly to Him, which I think is awesome. I don't know how every person feels, but I would venture a guess that most people that are offended by "OMG" have similar reasons.