No, an RS Mod simply means they moderate the RS sections on here. Nothing to do with Player Mod status in-game. It is the same procedure as gaining moderator status for any other section.
Don't worry, I made the same mistake when I came here haha. I thought that maybe the verified that they were player moderators in game and if nothing else showed at least a bit of maturity. Obviously not the case however xD
RuneScape mods on Sythe take care about RuneScape section. They have nothing to do with status of player moderator ingame. Basicly on Sythe that doesnt matter what you achieve/who you are ingame. You can only sell what you achieved ingame/sell your skills (for example in fire cape services) here but you wont get any extra benefits for it
What zuzel is trying to elaborately say is no - you will not.
Player moderators technically shouldn't even be on sythe unless they do not trade. Last I heard Jagex monitors Pmods accounts very strictly too to prevent them from botting/cheating in other ways etc
Jagex used to monitor them to shit back in pre 2010 but now they don't give a shit. Provided you're not muting people randomly they honestly don't care what the P-Mods do.