Hi 1914etc.
Ann, did you check the links I gave?
Yep.
All those converters show that 15 July from 588 BC and 4 July from 568 BC correspond to the same day (13 or 19) and the same month (Tammuz, the 4th) in the Jewish calendar, which from what I know is similar to the Babylonian calendar.
And did you read that I used to make the same mistake in thinking the Hebrew and Babylonian calendars correspond exactly? It turns out they don't.
In 588, Nisan 1 began on the evening of April 3 (according to Parker & Dubberstein's well-researched tables). Therefore, July 15, 588 = month IV (Du'zu or Tammuz), day 15, 588.
In 568, Nisan 1 began on the evening of April 22 - a little late due to an extra month XII immediately before. Therefore, July 4, 568 = month III (Simanu or Sivan), day 15, 568.
Your Julian dates are referring to the lunar eclipses in those years. Since the Babylonian and Hebrew calendars were based on lunar phases, it would be impossible for an eclipse to occur on day 13 or 19 of a lunar month. They would have to occur on day 14 or 15 when the moon was full.