Battery saver mode active?

Stop! Do not ever pull off a battery cable with the engine running. That is a very effective way to destroy multiple computers. You have enough problems already with the design of your generator. It is common to go through four to six replacements in the life of the car. Due to their design, they develop huge voltage spikes that can damage the internal diodes and voltage regulator and interfere with computer sensor signals. The battery is the key component in damping and absorbing those spikes, but as they age and the lead flakes off the plates, they lose their ability to do that. The issue now is not that you might think you have a "cheap" battery. The issue of concern is the loose cable clamp. Take care of that first, then let's see what you have for symptoms.

The "test" you're thinking of doing was done in the 1960s by mechanics who didn't understand how these simple charging systems worked. The problem is with no battery in the system to help the voltage regulator do its thing, it is possible for system voltage to exceed 30 volts, especially if you raise engine speed. No computer will tolerate that. In addition, if you could look at the output voltage waveform from the generator, you'd see it's not a nice smooth flat line at roughly 14.5 volts. It's actually three-phase output with peaks and dips, with the average being that 14.5 volts. Without the battery to smooth that out, some regulators respond to the dips, and thinking system voltage is too low, they bump it up by making the generator work harder. That's where system voltage can go too high. Some regulators respond to the high points and try to cut that down, with no success. Your engine could stall with a perfectly fine generator, or it could keep running with a defective generator, when you disconnect the battery. That test is pointless and doesn't prove either way that the generator is good or not. There's other ways to do the tests, and there's multiple ways a generator can fail. The only way to know for sure is with a professional load test. That takes just a few minutes, but you can start the testing yourself. Measure the battery voltage with the engine running. It must be between 13.75 and 14.75 volts. If that is okay, it only means it is okay to perform the rest of the tests. Those include "full-load output current" and "ripple voltage". I'll go into more detail later as to what those mean.

I'm here almost every day around this time, so don't panic if it takes a few hours for me to reply. Whenever you're ready to continue, let me know.

Dandy. The standard generator for your model is a 150-amp unit, which is pretty big. Start by measuring the charging voltage at the battery, with the engine running. It should be between 13.75 and 14.75 volts. If it is, I would continue with the professional system tests. Once the tester is connected, the tests take less than half a minute. What I'm interested in is the "full-load output current" test, and "ripple" voltage. All AC generators have at least six "diodes". Those are one-way valves for electrical current flow. In GM's generators redesigned for the 1987 model year, failure of these diodes is very common. When one fails, you will lose exactly two-thirds of the output current capacity, and ripple voltage will be very high. During your load test, output current is going to be pretty close to 150 amps, 50 amps, or 0 amps. Those are the only possibilities for a 150-amp generator. We want to see 150 amps.

If output current is good, indicating no failed diodes, ripple voltage will be low. There are a few testers that make a paper printout and actually list a value for ripple voltage, but most just indicate it as "high" or "low" with a series of flashing lights.

If the charging system tests okay and you still have symptoms, such as a run-down battery overnight, check the condition of the battery first. Most shops test it as part of the charging system test. The last thing is to test for a battery drain. To verify that is running the battery down overnight, disconnect the battery's negative cable while the vehicle sits. Reconnect it next morning, and if it cranks the engine just fine, we will have to pursue finding what is causing the drain.