The Spirit of the Terminator was apparently the soul or essence of one (or all) of the Terminators.
It was first invoked by Kim Richards in the Yogscast video, "GTA 5 Online - Driving to the moon (GTA Custom Games)". She invoked the Spirit with the plea, "Please, Spirit of the Terminator, grant me this."
The Spirit of the Terminator was a fannishinvention. As such, it has no direct analog in American English. Ones assumes the Spirit to be as ruthless and deadly as its onscreen counterpart.