My mistake - I'm to blame.

This slang term originated in about 1970. At that time, i.e. pre the widespread use of the Internet, slang terms often circulated at street level for many years before being adopted by anyone who felt inclined to write them down. That's clearly not the case any longer of course and any word or phrase that is widely known is dateable quite precisely via website logs.

The first citation in print is C. Wielgus and A. Wolff's, 'Back-in-your-face Guide to Pick-up Basketball', 1986:

"My bad, an expression of contrition uttered after making a bad pass or missing an opponent."

Shakespeare used the term with something like the current meaning, in his Sonnet 112:

Your love and pity doth the impression fill
Which vulgar scandal stamp'd upon my brow;
For what care I who calls me well or ill,
So you o'er-green my bad, my good allow?

That's clearly just coincidence, and it's hardly surprising that such a fragmentary phrase would appear in a large body of work like Shakespeare's. It's also a world away from pick-up basketball, which is an informal street sport where players frequently call out to each other (trash talking), and is a well-known source of street lang.

'My bad' came into widespread popular use in the mid to late-1990s in the USA via the 1995 movie “Clueless”. This starred Alicia Silverstone and contains what seems to have been the first use of the phrase in the mainstream media. The 1994 'Green revision pages' for the movie script has a scene with Alicia Silverstone's character learning to drive:

"Cher swerves - to avoid killing a person on a bicycle. Cher: Whoops, my bad."

Although a street term, it is virtually synonymous with the earlier Latin phrase, 'mea culpa'. It doubtless has as little of a direct descent from this as it does from Shakespeare's Sonnet 112.

'My bad' has gained that unequivocal accolade - imitation. In REM's 2004 song 'Leaving New York' there is this verse, which as you see includes 'my proud':

You might have laughed if I told you
You might have hidden a frown
You might have succeeded in changing me
I might have been turned around
It's easier to leave than to be left behind
Leaving was never my proud
Leaving New York, never easy
I saw the light fading out

The Doonesbury cartoon strip for 14th June 2006 included this:

"Okay, I'm bitter that I have to support myself! There I said it! My brave."