Even though you identified a problem, you are not checking correctly. The proper way to check charging voltage is post-to-post at the battery, and work toward the alternator if the voltage is low.

You can confirm alternator output by measuring at its output stud and alternator case. If voltage is 14+, move the negative probe to the block as this is the most likely point of loss on your car. The rubber mounted alternator needs a good cable between its case and the engine. Do not rely on visual inspection for this. If the voltage drop is on the other side - you can detect it by measuring between the output post and battery + post, working toward the alternator until you find voltage jumps exceeding 0.2V.