Asking other players to report or otherwise threatening with reports is considered negative behavior that may be punished. One game of this wouldn't result in a chat restriction, but a consistent history of this sort of behavior could. Punishment is calculated per match not per report, so all you need to do is report them silently and let the other players decide on their own what should be reported.
Also, your punishment has no bearing on whether your friend will be punished or not. If your friend continues his negative behavior then he may be subject to a chat restriction or ban later as well, but most likely the reason you were punished this time and he wasn't would be that he isn't consistently toxic.
If you want to discuss your punishment with the community I'd suggest posting your whole reform card here. If you want to appeal it then you need to submit a Support Ticket, but Riot will not remove a punishment that was not done in error.