Except neither study is talking about the most the most common form of suicide. They're both reporting higher rates of suicide among gun owners.

Me owning a gun doesn’t increase my likelihood to die

That's literally what both studies are saying. From the first one:

Men who owned handguns were eight times more likely than men who didn’t to die of self-inflicted gunshot wounds. Women who owned handguns were more than 35 times more likely than women who didn't to kill themselves with a gun.

From the second one:

in Wyoming, where 63 percent of households reported owning guns—rates of suicide were higher. The inverse was also true: where gun ownership was less common, suicide rates were also lower.

Also from the second source,

Studies show that most attempters act on impulse, in moments of panic or despair. Once the acute feelings ease, 90 percent do not go on to die by suicide.