We just had our Gen3 connector installed by a "Tesla certified" electrician. They did it right, but had no idea how to commission it, so I did that myself via the TeslaPros app. Once that was done, I added it to my house via the standard Tesla App. There were a few bumps along the way, but I got it done and now I don't see any lights I don't expect. Blue when plugged into the vehicle but not charging, green streaming while charging, and single solid green when not plugged in. Connector also shows up in the house view as either plugged in or idle, and when charging it shows how much current is going into the car. Oh, I should probably mention we have a pair of PowerWalls, a Tesla gateway, and solar panels, which may be why we have a house view in addition to the car view in the Tesla App.
As has been noted above, we could force the green pulsing light by holding down the handle button long enough, but we don't do that.