Okay, so that does sound like it is in the tendon if it is central and 3 inches up from the bottom of the foot. Can you feel a bump in the tendon? It is likely sore if there.
And just wanted to mention if you wear orthotics for pronation, barefoot running may be a painful prospect for you.
How tight are your calf muscles? Do you stretch? Do you foam roll? Think of the tendon as the weak point in the muscle tendon relationship. When the muscle is tight you can feel pain in the muscle, but often you feel pain at the attachment to bone point, the tendon, or musculotendonous junction. If you relieve the muscle tightness, the excess overstrain on the tendon reduces and the pain reduces. Simplistic, but the way it does work.
I've seen it many times that active people don't foam roll and stretch and their muscles are in an extremely tight state.
Getting physio to help relieve the inflammation and to set you up on a good program is ideal.