It used to be the case in the US that women attempted suicide at a higher rate than men, but men succeed at a higher rate. The difference was said to be the method, and of course guns were a favored method with men.
It is still the case that women attempted suicides is higher and the results a lot lower, but as far as I remember in most cases for women it was just a way to cry for help, not to actually do it, while men were on the other side. I cannot find a link right now, I think it was a video of a psychology class.
I think for most it's a cry for help, but some people die before they get the help because their method is too effective.
Of course there are also people who have carefully considered their options and decided they really want to die, but I suspect that's a very different group.
Intention of success is hard to know... What we do know is women use less lethal methods like pills, rather guys who use more lethal methods like firearms.
You can do that by comparing the rates in various states against the rate of firearm ownership. There does seem to be a positive correlation between firearm ownership and suicide in the United States, although other correlations are equally present and might be causative as well. Rural areas tend to to have high suicide rates[0], and tend to both be more poor and have more firearms than urban areas per capita.
Globally it's more complicated, as there are plenty of countries with fewer firearms than the US and more suicides, as well as countries with fewer firearms and fewer suicides. Material conditions (poverty, etc.), prevailing attitudes around substance use, hard to describe social well being factors, and stigma/quality of mental health care seem to be factors.
For my part, I think that easy access to a means of suicide, such as firearms, is one part of the equation, but not the dominant part. The dominant part is the prevailing experience in the population as a whole, as a population that feels like it's lost hope is more likely to attempt suicide on average than one that has not. But it does seem like wide spread misery and access to an easy and quick means of suicide is a very bad combination.
Doing the comparison between South Korea and USA, restricting firearm access look to be increasing suicide rate. Of course this is not true, just an extreme outlier, the point is that cultural differences have a much stronger impact than access to a particular method.