I'm pretty sure that Houdini's fictitious account of how he got the bullet in his hand was in a short story called "A One-Night Engagement" which was published, as I recall, in 1904 in, I think, Wide World Magazine. I reprinted the story in "Houdini's Strange Tales" along with a bit of history.
Houdini wrote to Bess that she should pick up Wide World Magazine to read how "I shot or was shot."
Nobody knows how he really got the bullet in his hand.
Christopher wrote that he had the bullet in his hand when he rejoined the family in New York after his runaway days.