first of all, you must remove the negative cable once the car is running, not the right way, you need to remove the positive cable.
if you do remove the cable once the engine is running, if the car dies, then the alternator is kaput. if not, but reading is 13.6v, then something is not right.
and i know why..
buy me a drink and i will tell you.
just joking, if the charge is not good back to the battery, that means the connector spiltter from the battery cable to the fuse, alternator and starter is corroded or got rust/dirt. if bad, then source for a new one or renew the whole cable.
but you must check with the volt meter between 2 points, one at the alternator output, and the other at battery cable spiltter. see the difference, should be reading at 14.2 above, not less. if at the alternator, then change it, not good recon model. if not then it's the cable.
btw, the above two alternator is the same, one older one newer. just make sure you have the pin outputs same.
cheers..