I think people lying about rape harms the legitimacy and impact of the word more than people using it in a sports context.

People understand what it is and how serious it is

I really doubt that people understand how serious killing others are given the split reactions to the following situations:

• War
• School Shootings
• Police Shootings
• Murder in film, video games, art, literature.
• Number of killings world wide

People are so use to hearing about killing in media that it has diminished effect, but when people hear about rape, there is a riot and everyone agrees what it is and how bad it is.

Out of the factors that make up "Rape culture" :

• Blaming the victim (“She asked for it!”)
• Trivializing sexual assault (“Boys will be boys!”)
• Sexually explicit jokes
• Tolerance of sexual harassment
• Inflating false rape report statistics
• Publicly scrutinizing a victim’s dress, mental state, motives, and history
• Gratuitous gendered violence in movies and television
• Defining “manhood” as dominant and sexually aggressive
• Defining “womanhood” as submissive and sexually passive
• Pressure on men to “score”
• Pressure on women to not appear “cold”
• Assuming only promiscuous women get raped
• Assuming that men don’t get raped or that only “weak” men get raped
• Refusing to take rape accusations seriously
• Teaching women to avoid getting raped instead of teaching men not to rape

only 4 of them are critical or legitimate. Adding any of the crossed out items loosens the meaning of rape since they do not imply an acceptance of rape where rape is defined sexual penetration without consent.

If you define murder culture as loosely as rape culture, the US would have a significantly larger problem with "murder culture."