WAYNE, N.J.—After several starts and stops, Brandon Jacobs finally settled on the line to start his own story: "I'm a normal person."

For six years in the NFL now, and several in college before then, someone else has been writing the story of Brandon Jacobs, the bull of a running back for the New York Giants, a 6-foot-4, 265-pound "thumper," as coach Tom Coughlin called him last week.

Today, the most common telling of Mr. Jacobs's story is that he's a lineman-sized malcontent, a petulant little boy who had his ball taken away and so is sulking. After rushing for more than 1,000 yards in consecutive seasons and playing hurt all of last season, Mr. Jacobs this season was supplanted as the Giants' primary tailback by Ahmad Bradshaw.

His helmet-flinging incident in Indianapolis in Week Two, which he says was accidental, and the subsequent reported trade request, which he insistently denies making, wraps up the caricature: Mr. Jacobs is a sometimes forceful, sometimes moody emotional liability.