I think it's something along the lines of the latter--men are more likely to use violent means that are more effective where women are more likely to use means

I think it's something along the lines of the latter--men are more likely to use violent means that are more effective where women are more likely to use means such as pills, which are less effective (but can still lead to permanent damage). Additionally, violent means lead to instant death more often so there's less opportunity to back out or for someone else to intervene. So there are certainly gender differences, but it's not clear that it's a difference of intent.