I created this question a while ago and received some help that led me in the right direction but did not fully answer my question as it stands. After following up with the method of that first answer, I wrote a new question which is here.
The new question is clearly different, in that it directly addresses two algorithms for weighted quantiles, and why they don't match with Stata, whereas the first question doesn't deal with specific algorithms and doesn't address the distinction between equal number and equal weighting quantiles.
I agree I could have titled the second question differently, to distinguish it better, but it is definitely a distinct question that I feel meets the FAQ guidelines.
I was about to open a bounty on it this morning when I noticed it has been closed as an exact duplicate. May I please have the question re-opened?