However, once I've refilled the reservoir, turned the wheel to full lock positions about 8 times, and turned on the engine, there's a loud, god-awful noise coming from the steering, I assume the pump. When turned the engine off and checked the reservoir, there was still fluid in it, although it had gone down some, but the fluid in the reservoir is very frothy (i.e. it has lots of visible small bubbles).
Noise is air in the pump. Turning the wheel lock to lock on filling doesn't help. Only the pump can get fluid into the rack.
You want to bleed the air from the pump and lines by turning on the engine for just a few seconds (off before you hear a noise), then refill the reservoir, repeat.