> men who owned handguns were eight times more likely to die of self-inflicted gunshot wounds, and women who owned handguns were more than 35 times more likely to kill themselves with a gun

Emphasis mine. This says nothing about suicide rates, just suicide by gun rates.

Yes, people who own guns are more likely to kill themselves with a gun. This is obvious.

People who own stairs are more likely to fall down the stairs. People who have a dog are more likely to be bitten by a dog. Etc, etc.