Two important Arabic words to learn.
TAKFIR goes with Allahu Akbar. It is the call to scream Allahu Akbar in unison.
TAKBIR means declaring a less than perfect Muslim (according to the jihadi with the gun) an apostate, as an excuse to torture, rape and murder him and his family.
I think it’s the other way around. Takbir comes from akbar (great) as in allah akbar. Takfir comes from kafir (infidel or unbeliever).