Let's look at how the charging system works:
The alternator charges the battery directly thru a cable that (I am 99% sure) goes right from the alternator to the "hot" side of the starter solenoid. This connects to the cable that is the BIG positive battery cable that then goes right up to the battery. There MUST be a good ground path from the alternator to the battery as well, that is the circuit: From alternator, to battery via cables, then from battery thru it's ground cable, thru the chassis, thru the jumper from chassis to engine block, then from the block thru the alternator mounting bolts and into the frame of the alternator.

These alternators use an internal regulator, and the ONLY way they "know" what the battery voltage is, is via a LOW resistance path as described above. Any corrosion or "green" (that is, formerly wet and now corroded) connections will mess up the alternator's ability to charge.

Power for everything ELSE in the car comes from the battery, down the SMALLER cable from the Positive Battery post that goes directly to the four-fuse little box just ahead of the battery. Power is distributed to the entire car thru the three fuses (one position is unused) in that box. There are many paths taken from that box, section 8W of the manual diagrams this for you.

Perhaps the battery IS being charged just fine. Perhaps you have a poor feed of power to the car.

I would check for corroded battery posts/cable clamps and a corroded battery ground cable (look over CLOSELY the lug and screw where it attaches to the fender/etc., just ahead of the battery). ALSO, there is (somewhere, never found it myself) a heavy ground strap from the engine block to the chassis. On my car, I suspected this cable to be bad when I bought the car. I made up a piece of Number 6 insulated ground wire with two soldered lugs on it. One attaches to the negative battery post clamp, the other goes to a bolt on the engine block (can't remember which bolt I used and I'm 700 miles from home this week).

It was good for you to measure the alternator output. The next step WHILE IT IS DOING IT, would be to measure battery voltage. After that, go measure the voltage at, say, the cigarette lighter, or one of the fuses in the various fuse boxes. SOMEWHERE, you are seeing a drop in voltage that is "offending" or "confusing" one or more modules in the car - that is why the abs light came on.

If you come back telling us that the battery is at 13.5 or more but you only see, say, 11 volts on the cigarette lighter, or on some/all of the fuses in one or more fuse boxes, we can then start to trace where the problem is.

The air pump probably does draw 10 or 20 amps, but the car is DESIGNED TO SUPPORT THAT. If the pump coming on causes problems, it's not likely the pump is drawing too much, but instead that the system is in trouble and cant' support the normal load.

ONE MORE THING, My car's instrument lights flicker ONCE, every once in a while. I suspect I am soon to experience trouble for myself. It sounds to me like SOMETHING is turning on in my car and the system is barely handling it. But maybe not.....?