The second is the actual Tenseigan dojutsu activated by Toneri, and Hamura before him. After Toneri transplants Hanabi's Byakugan, they react to his Ōtsutsuki chakra and transform into Tenseigan. Contrary to early reports and interpretations, this apparantly has nothing to do with the countless Byakugan sealed in the Energy Vessel altar and Toneri's Tenseigan isn't channeling their power (rather, the Energy Vessel housing Hamura's Tenseigan is, hence the confusion).
So, not unlike the Rinnegan, it's actually awakened by Hamura's two bloodlines uniting (Toneri's Ōtsutsuki chakra and Hanabi's pure Byakugan). Hanabi's Byakugan are especially pure in that, as an heiress to the Hyuga Clan's Main Branch, her bloodline is undiluted. The same is true for Hinata and Hiashi.
