Yes, you are.
I'll break it down for you:
WiiConnect24 just connects online. The Standby Connection (Don't confuse this with the normal standby that all electronics these days are in when you turn them off) keeps that connection active 24 hours a day. You cannot have the Standby Connection on without having WC24 on, but you can keep WC24 on without the Standby Connection being on so that it'll only get online when you have the Wii on. The problem is that the Standby Connection is keeping the Wii on all hours of the day without running the fan, thus making the Wii hotter than if the Standby Connection were turned off.
I just went to check it after I posted and I see what you mean now. Looks like Nintendo fucked up with it, you're not really in Standby mode since the damn thing is still running hot.
Edit : But still, red light=standby mode. Its best to either leave WiiConnect24 off or like you and others said to have it on but turn Standby mode off without having to unplug the power cord. Whatever you do, just make sure you don't have the orange light or it'll run hot and your system.