It's a great question, and in my humble opinion, the answer is:
Phalanx (plural: phalanges), (also Cambridge and MedicineNet) is the term for one of the **bones** (only) in the fingers or toes and not for all the "meat" around it. The Google images search for phalanges is quite convincing.
Someone on < might know a word for the whole bone + "meat" part of the finger.