Even within a given category of method of suicide, men have a greater success rate. Places where the most efficacious method of killing yourself (guns) are banned still maintain a ~3:1 male:female suicide death ratio.

A naive read on suicide attempt:success rates suggest that men are literally 10x as effective of women in achieving their aims. That's the kind of ratio that suggests that there's something big not being captured in the statistics, and in that case it's important to focus on the more objective measure (a death by suicide) than something relatively subjective (a suicide attempt)