Interesting that you use the phrase "quality vs quantity". Normally, I think of precision being on the side of "quality". As in, you carefully make a thing to meet a certain standard instead of making a lot of them.

In the case of "qualitative vs quantitative", despite having the same roots, I think of precision being on the side of the quantitative --with qualitative work answering yes/no questions and quantitative yielding precise measurements.

Yes, the latter is what is implied. "Quality" has more than one meaning, and "observed characteristic" is what is meant, not "goodness".