New Year's is determined by the Gregorian calendar while Chinese New Year is determined by the lunar calendar.

Because back then, the Koreans used a different calendar, which went by the moon instead of the one that goes by the sun.

Your coin is dated 1959 in U.S. terms, or 4292 - from the Korean calendar!

the Gregorian calendar

No, the 2012 calendar has its dates on the same days of the week at the 1984 calendar and the 2040 calendar. The 2000 calendar is likewise the same as the 1972 calendar and the 2028 calendar.