The pins I've circled in red here are the "Control Pilot" pins, and my working theory is that they're how the 12V battery charge is leaking out of the Clarity. (Note that the picture has a Tesla plug, not a Tesla receptacle; on your adapter, you'll have a Tesla receptacle so the pin will be on the other side. For further reference, both pins should be on the left side of their respective sockets when the adapter is plugged in.)
I think the Tesla adapter probably just connects those pins together with a wire, but the third-party adapter has a diode which only allows current to flow into the car and not back out again. If so, you could use the continuity mode on a multimeter (the one where it beeps when you touch the probes) on those pins in Tesla adapter, and it would beep, while on the third-party one it wouldn't. Furthermore, in diode test mode, the multimeter should show a voltage (probably 0.6-0.7 volts or so, possibly less) when you have the negative lead connected to the J1722 CP pin and the positive lead connected to the Tesla CP pin, whereas the meter should not show anything if the leads are the other way around.
If that's the case, I'll probably try to DIY a fix for my Clarity; no promises I'll actually succeed, but I'll be happy to share progess one way or the other.