"…I have read a Lucile's testimony about lowering of the lifeboat No. 1. She said that the people were fighting and trying to get in to that boat and captain Smith shot some of the passengers…"
That isn’t from Lucile’s testimony at all. That is from an erroneous account in the New York press, one that was widely syndicated. She never actually gave that interview, so don’t rely on it. There were a number of fictitious stories attributed to her in the wake of the disaster because she was a known name and "good copy."
"…And Cosmo was describing the scene very differently."
Cosmo never gave any interviews, unless you count an expletive or two that he hurled at obnoxious reporters who virtually assaulted him on arriving in England for the Titanic Inquiry.
"…I had always been careful about LDG statements…"
You need to be more careful of newspaper accounts attributed to her. The only authorized interview she gave after the sinking was with the New York American, and even it was marred by sensational additions made by the reporter she spoke with by telephone.
"…At a certain point she appears to become nervous and gives bad and confused answers to the questions…"
If you’re referring to Lucile’s testimony at the Inquiry, you are right. She was indeed nervous at the hearings — and for good reason. She and Cosmo had been heavily coached in their evidence by their lawyers. Her testimony shows she was distracted, often glancing over at her attorneys for direction. At one point a cross-examiner called her down for this.
"…I don't know, but I can just say I never saw any records of Captain Smith using his gun that night…"
If he did, Lucile didn’t witness it. Again, the interview with her that claimed she saw Smith shoot people never took place.