TMZ reported hours after the shooting, a well-placed source who had spoken with Secret Service told us ... Trump was hit by pieces of glass that had shattered during the gunfire.

The narrative changed almost immediately when the official word came out ... that Trump was grazed in the ear by an errant bullet, and had he not turned his head he would have been dead.

Olbermann weighed in on X ... "In brief: Ronny Jackson isn't a doctor. Which is perfect, because Trump wasn't hit by a bullet."

Olbermann is referring to our story ... that now-Congressman Ronny Jackson's medical license is not current.

Jackson evaluated Trump Saturday, saying Trump was hit by a bullet in the right ear with Thomas Crooks' high-powered rifle.

Jackson said the bullet carved a 2 centimeter-wide gash that went all the way down to the cartilage surface of the ear. Jackson added, "There was initially significant bleeding, followed by marked swelling of the entire upper ear. The swelling has since resolved, and the wound is beginning to granulate and heal properly."

Then Jackson went on ... saying "Given the broad and blunt nature of the wound itself, no sutures were required."