In fact, many Asian countries use lunisolar calendars, which basically follow the moon for the months but add an extra month every few years so the seasons don't drift.

As these calendars also rely on time zones for date calculation, there are rare occasions where the New Year start date differs by an entire month between 2 countries.

If that's a sole problem, it should be called "Chinese-Japanese-Korean-whateverelse new year" instead. Maybe "East Asian new year" for short. (Not that there are absolutely no discrepancies within them, but they are so similar enough that new year's day almost always coincide.)