It's great to see you here again, Alleymom! Thanks for drawing attention to the 'lunar threes' again.
Notice how endnote 18a goes on to cast doubt on the reliability of these measurements, despite referencing the same work Alleymom cited - Under One Sky!
18a. These time intervals ("lunar threes") are the measurement of time from, for example, sunset to moonset on the first day of the month and during two other periods later in the month. Scholars have tied these time measurements to calendar dates. ("The Earliest Datable Observation of the Aurora Borealis," by F. R. Stephenson and David M. Willis, in Under One Sky-Astronomy and Mathematics in the Ancient Near East, edited by John M. Steele and Annette Imhausen, published 2002, pages 420-428) For ancient observers to measure this period required some sort of clock. Such measurements were not reliable. (Archimedes, Volume 4, New Studies in the History and Philosophy of Science and Technology, "Observations and Predictions of Eclipse Times by Early Astronomers," by John M. Steele, published 2000, pages 65-66) On the other hand, calculating the position of the moon in relation to other celestial bodies was done with greater certainty.
Regarding the 'lunar three' measurements' 'unreliability,' Steele's work is used as support (the pages can be viewed on Google books).
The trouble is, Steele was specifically referring to the timings of eclipses - far longer events and therefore prone to a little more error. But even so, he says that these water clocks have an accuracy of about 8½%, so e.g. a 2h 40m eclipse would have an accuracy of just under 24 minutes (see p. 66).
Typically, 'lunar threes,' 'lunar fours' and 'lunar sixes' are far smaller timings and are generally reliable - in the case of VAT 4956, to the order of 1°, according to Stephenson and Willis, as was mentioned above. As Alleymom has said before, the 'lunar three' timings on VAT 4956 would in themselves knock 588/7 BCE as a possible alternative year out of the running, and this will be why they are dismissed by the WT article writer.