Alternator charging problems?

From the description you have more than one problem. First I would get the battery load tested to find out if it is still okay, they don't like going flat and being recharged repeatedly. Next you need to test the replacement alternator. From your description it sounds like it is either bad or the control circuitry has a problem. A few things to look at. First does the battery light come on with the bulb test function does it stay on with the engine running?
Next take a test light and unplug the alternator, turn the key on but don't start the engine. Use the test light and check all three of the pins in the harness connection for battery voltage. With the key on they should all show battery voltage because they are being fed power through the fuses and the ignition switch. If any do not show voltage that circuit is the problem. If they all test good then it is likely your alternator is the problem. It wouldn't be the first time a new part was bad out of the box and it has been getting more frequent.

https://www.2carpros.com/articles/how-to-check-a-car-alternator

As for a higher output alternator, they put out higher amperage, not voltage. They are good if you have a big stereo or lots of electronics to power. Otherwise they are a waste of money because you don't need the extra amperage.

Really dim isn't a good sign, what pin and wire color was it? The white wire is the sense control, it tells the alternator the battery voltage and controls the charge rate from that voltage. It feeds off the fusible link assembly directly from the battery. The red wire with blue stripe is the power feed to the regulator from the ignition switch, it comes out of a fuse, through the ignition switch and to the alternator. Both should be full brightness unless they have a problem. The yellow wire feeds the indicator light and does two things. If the alternator is working it will go out as the voltage from the alternator equals the voltage being fed to the bulb from the instrument cluster. It becomes a ground for that lamp if the alternator stops working. The bulb test is the lights on the dash like an oil, temperature, alternator, check engine and related ones that come on when you turn on the key then go out.

It acted this way with the old alternator as well as the new one correct? Take a voltmeter and test the voltage output right at the alternators battery post, then at the battery. Engine running at about 1,500-2,000 rpm's. Are they the same number and what is it? Now do the same with the battery post from post to engine ground and post to alternator case. The number you posted suggests the alternator is not doing anything. Try doing the same voltage tests on the wires with the connector plugged in and the engine running, what does the voltage show now?

Okay, if I understand what you posted you have a wiring problem. Looks like a voltage drop between the alternator and the battery, likely from one of the grounds being corroded. The 11.75 volts suggests that. From the battery post to the case should read full battery voltage. As a test you could run a jumper wire from the case of the alternator to the negative battery post, that would bypass all the other grounds, Do that with the engine running at 1500 or so and measure the battery voltage itself, then shut it off and measure again. Any change? Being a 2001 it doesn't have the modern controls that throttle the alternator based on current usage, it should put out charging voltage all the way down to idle.

From the numbers you have a bad connection. The alternator is charging as shown by the 13.93 and 14.00 volts in your second set. But when you apply a load as in the 12.44 idle under load, the voltage drops off. That is a classic sign of a poor connection, AKA voltage drop. For example say I take that meter and connect it to a single small strand of wire, with a 12 volt battery at one end, I will read 12 volts through that wire because the meter doesn't apply any load to the circuit. Now if I were to connect a 12 volt bulb in there and then measure it, I may see only 10 volts or so on that wire because that single strand cannot carry enough current to keep the voltage up. Same thing happens on a vehicle, a bad connection or corroded wire acts like a resistor and reduces the voltage but only once the load exceeds the ability of the connection to carry the voltage.
To test where the issue is, Take your meter set on volts. Now connect it between the positive battery terminal and the battery stud on the alternator with the engine running and the headlights on. Next take the meter and connect it between the negative battery terminal and the case of the alternator or a good bare spot on the engine. In both cases with very good wires and connections you in theory should see no voltage on the meter as all of the current should be in the heavy cables. In the real world you will see a tiny voltage like.10 or less because the cables do have some resistance. However if you see a voltage over one half a volt or higher, that cable or connection is bad. From the numbers you have posted I think you will see closer to a volt on one or the other which means a very bad connection. That drop would get higher the more of a load on the circuit as well. So say you are running the car and have the A/C on and the headlights with the interior blower on high, the voltage drop could easily be 3-4 volts which would drain the battery and cause your problem.
We will find it though.

No, leave the cables connected while you test. The idea is to create two paths for the voltage to go. If the battery cables, connections and wires are good all of the power should go through the heavier cables with only a tiny voltage on the meter. However if a connection connector or cable is failing the meter will read a higher voltage as the power travels through it and not the main cable.
If you have new cables connected on top of the old ones, it is more likely to be a bad connection between the old cable and the parts. I would remove them, clean up the areas they connect then try it. You would be amazed how little it takes to stop a system from charging.

No set it to volts. You are testing similar to this image except to the alternator instead of the starter. Say you have a fully charged battery at 12.6 volts. You start the engine and it now shows 13.6 volts, that is charging. With the meter connected as in the picture between the battery positive and alternator output posts and the cable connected with a load like the lights on, you should see a very low voltage, no higher than about.2 or so at the most if the cable and connections are good. If you them do the same on the negative side you should see a very low voltage. In a perfect world it would read zero, however nothing is perfect when it comes to cars. I think you will see a higher number on the positive side though as with the amp draw there should have also been a low like.05 that you had on the negative side if all of the current was going through the cable as it should.
If you want to see the idea just search for voltage drop testing, there are a lot of videos out there on it, from basic to explaining it down to electron flow.

Is that with a load on and idle? Try revving it up some. Then try the ground side. If it still reads low then it almost has to be on the sensing side of the system, or the feed into the vehicle. You can use the same testing method between the various connections as well. That is one advantage to voltage drop testing, you can do it with the circuits under power without removing a connection.

Hello my name is Dave.

First thing I would do is test the battery, new or not it can still be faulty and keep the system from charging properly. I have attached a link to our tutorial to help you with this process.

https://www.2carpros.com/articles/car-battery-load-test

This can also keep the alternator from being able to provide enough amperage to fully energize the headlights. If the battery tests good I would suggest making sure it is fully charged and then test alternator. I have attached a tutorial to help you with this process.

https://www.2carpros.com/articles/how-to-check-a-car-alternator