I agree with you, you cannot use the drug itself to determine the type of anesthesia that is used.
With regard to CPT code assignment, there are some good articles in CPT Assistant that might help with this, for example the article from January 2018 on Fracture and dislocation restorative services, which states that "For purposes of coding, use of local anesthetics is not considered "anesthesia" because local infiltration is always part of the surgical package.
Some of the CPT codes specify in their descriptions what level of anesthesia meets that requirement (e.g. 21073 which states "requiring an anesthesia service (ie, general or monitored anesthesia care)" or 69424 - Ventilating tube removal requiring general anesthesia) whereas others simply state "with/under/requiring anesthesia" (e.g. 20693, 20694), which per my understanding would imply any anesthesia other than local.
I don't have access at the moment to the latest editions so there may some guidance more current than this, but hopefully this helps some.