Battery light/charging issue

To start, I've been searching the net for an answer to my problem for about a week and the best answers have been on this forum though not quite solving the problem.

The car is my wife's '03 V6 convertible Mustang. It's bone stock, in amazing condition with 190k miles and never been hacked on when service has been needed.
Last week she said the battery light had been coming on but not staying on. Starting and driving ok. Thought it be ok until the weekend, Friday night it died on her close to the house. Pulled the alternator, went to autozone to have it checked. He said it failed the test but I was suspicious of the guy and the test but whatever, I bought a rebuilt.

The problem is that it starts, charges at 14.4v -14.2v then anywhere from a few minutes idling in the driveway to a trip to the store a mile away the light comes back on. When the light comes on it is still charging at 14+volts but shortly thereafter the alternator will stop charging and not start charging again. It still has voltage at the 2-wire connector on the alternator. (battery volts and ~1.75v on the other). If I pull fuse 1.20 in the fuse box the alt kicks back on.
I've checked wires, terminals, checked for voltage drops and all seems good. I'm looking for any ideas and procedures for troubleshooting.

Thanks in advance all.

The cables appear good and I did pull them to clean them, and the battery posts, with my cleaner. They are still stock though so I may replace them just because. I didn't pull the battery and have it checked because the car starts quick and has over 12 volts but I will this weekend. The grounds seem good, I see no difference in voltage between battery ground and chassis ground. Same between alternator B+ and battery +.

I was thinking bad alternator also.
The strange thing is that it starts and charges. It's only after a bit of time that the light comes on and then finally there is no charging. Almost like it is heat related. Loading it with the A/C and lights doesn't make much difference while it's still charging. 14.1 volts.

I'll pull the battery and alt this weekend and take them to a different store to be checked. If they check ok, I'll start replacing terminals and cables.

So to update everything thus far:
I did ohm everything as you laid out a few days ago not seeing more than a tenth or two of one ohm. I did checked the voltage between battery +, battery terminal on the fuse box, and the battery + post on the alternator. Not more than 2 or 3 hundredths of a volt drop. You can start to see my frustration.

Saturday, I took the battery to a different store (o'reilly's) where they did a long test on it. It showed to be good.
I took the alternator back to AZ where it checked ok again. The guy there talked me into bringing the battery in to him so he could be sure. I think the O'reilly test was more complete but whatever. He said the battery is marginal and just met the requirements of the car. In other words, he talked me into a new battery. I'm ok with it since it was a cheap battery and more than 5 years old anyway.
Get the battery home installed and go for a shake down cruise. A mile from the house the light comes back on, goes off then back on again. I head straight to the store for them to do an 'on the car test', it tests bad. He says bring it back and we'll swap it out.

This morning I pull the alternator off again, take it in, and it tests good. They are willing to do a "courtesy exchange" but are 100% sure I have a bad wire coming from the alternator because "if it tests good in the store but bad on the car, the car has a bad wire". I ordered a replacement the lead before I left to save time just in case.

Put the second alternator on and went on a road test. For an hour! Stop and go traffic, out to the freeway and back home. No light! The wife then took it to church and to the store. No light! I think it's fixed.
It really sucks when you can't trust the part that just cost you $200.

wmburns I really appreciate your responses and suggestions.
I'll report back if there anything else happens.