That's an excellent find, Don that I think definitely explains the discrepancy.
One thing I note is that the months of the ancient Greek Calendar had rotating numbers of days, so any fixed "x of Gamelion" date will always translate as a different date on the Julian calendar depending on your starting point. Since, as Don just demonstrated, Epicurus was, in fact, attesting to the Eikas-aligned celebration of his birthday on "the [earlier] tenth" it would be appropriate to celebrate it in correspondence with our own Eikas-day celebrations.
At the same time, we could approach this holiday as a "moving feast" like the Christian celebration Easter which can fall anywhere from March 22 and April 25 depending on the year and the movements of the moon. Celebrating Epicurus' birthday on the "proper", "moving" date could be a fun, educational exercise we can celebrate every year.
Based on the Calendar Don provided (above), I believe we are in the 2nd year of the 700th Olympiad, which started on June 29-30th 2022. Today, November 24th 2022 corresponds with the 148th day of the ancient Greek year, meaning "today" is the final day of Maimakterion. This year also seems to have a second month of Poseideon before Gamelion, so the "early 10th of Gamelion" (or Gamelion 20) falls 227 days after June 29-30th is February 11-12th 2023.
Last ancient Greek year (the 1st year of the 700th Olympiad; pictured above), the first month of Hekatombaion began on July 10-11th 2021, and the “early 10th of Gamelion” (or Gamelion 20) fell 197 days after July 10-11th. So, according to my calculations, last ancient Greek "year", Epicurus’ Birthday Eikas would have fallen on January 23, 2022.
In next corresponding ancient Greek year (the 3rd year of the 700th Olympiad; seen above), the first month of Hekatombaion begins on July 18-19th 2023, and the “early 10th of Gamelion” (or Gamelion 20) will fall 198 days after July 18-19th. So, next ancient Greek calendar year, Epicurus’ Birthday Eikas will fall on February 1-2nd, 2024.
So based on my calculations:
Epicurus’ Birthday Eikas 2022: January 23-24th
Epicurus’ Birthday Eikas 2023: February 10-11th
Epicurus’ Birthday Eikas 2024: February 1-2nd
I think it is therefore appropriate to celebrate it on one of two days:
1. January 20th, aligned with the modern celebration of the Twentiers
2. A "moving feast" like Easter, which will start Epicurus' Birthday Eikas (as I count) on February 10-11th 2023.
I am partial to celebrating the "moving feast". It provides an opportunity for modern Epicureans to gain a better understanding of the Attic calendar that organized Epicurus' days. It also reinforces our relative feeling of their holidays. There is a slightly different feel between contemporary cultures that start their "week" on Monday instead of Sunday. There is a different feel to "beginning" a year in Summer instead of mid-Winter. This helps us look at time from the perspective of Epicurus and friends. It also provides an opportunity to focus on Epicurus.
Or we could just do the 20th to make it easy. Easter's moving date always was kind of weird growing up.