The new menu screen does fail to open (accidental double click), it would make sense for it to failsafe to solo. Not a big deal though.
I don't think it matters what they are called as long as it is clear. I understand what Open means both independently and in the context of the other two options.
Cheating is clearly a problem that needs to be addressed, I'm not convinced renaming mode choices would be even partially effective in dealing with this.