I am not an astronomer, nor do I have an ephemeris handy to check this out, but since there is a 10-year "difference" here, and since eclipses occur every year, I don't see the debate ending any time soon.
Then there's the definition of "date". The calendar used at the time? The Julian Calendar? The Gregorian calendar? modern "adjusted" calendar?
What?
They were looking for a solar eclipse that would have been visible in Ithaca (not the "City of Evil" one!) which narrows the field considerably. And, if you had read the article, you would have noted that a number of planetary positions were figured into the mix.
As for the exact meaning of "April 16, 1178 B.C.", what's the problem there?
If my astronomical calculator is correct, there was a new moon on April 16, 1178 BC Julian.
Solar eclipses are very specific in their timing and their location. They don’t happen every year in the same place.
Not in the same place, and not necessarily total. For example, we're going to get a total eclipse here in the continential US in 2017, and our last one was in 1979.