State Department is in charge of these designation, but it often doesn't rise to the level of the Secretary of State until the lower departments have come to a conclusion. There was a debate within the State department over this between the counterterrorism bureau and the Africa desk.. These designations are not cost free and each side has it's arguments. The folks arguing against this felt it would raise the profile of a criminal group to a higher level which might help them recruit. Boko Haram were pretty much just in Nigeria at that time and a designation would essentially label them as a threat to US Nationals or the National Security of the US, giving them a higher status. Nigeria strongly did not want them to be designated a terrorist group for this very reason. because it felt doing so would strengthen the group.

Another issue was we were trying to get Nigeria to curb the abuses of its military. Designating Boko Haram might legitmize some of these abuses (which experts thought were helping Boko Haram recruit.)

The designation doesn't mean we were fighting against Boko Haram or helping Nigeria fight them.

If you want to see a summary of the arguments against the designation, you can see this 2012 letter from academics who studied the group

Note also from Wikileaks that Boko Haram declared allegiance to ISIS. This fits George Webb's narrative. https://wikileaks.org/berats-box/emailid/2495