It’s, scientifically, Chinese New Year.
And it’s a problem only for people who celebrate the day outside of China. In China, it’s formally known as 农历新年 (Agricultural Calendar New Year). And the holiday is known as 春节 (Spring Festival). But, traditionally, the Spring Festival falls on 立春 (the Beginning of Spring), a solar term, on Feb. 3, 4, or 5 of each Gregorian calendar year. But in most modern times, or after the founding of the Republic of China (1912-1949), the legal days-long Spring Festival have gradually come to start (most recently, or just a few years ago) on the last day of the agricultural calendar, or New Year’s Eve. So the two (the “Beginning of Spring” 立春 as a solar term and the “Spring Festival” 春节 as a holiday) can be days and even almost two weeks apart.
The calendar is also informally known as 阴历 (lunar calendar) in spoken Chinese. But scientifically, it’s a lunisolar calendar, not precisely lunar. The agricultural calendar, formerly and sometimes informally known as 夏历 (Xia Calendar, traditionally believed to be from the Xia dynasty, ca. 21th century – 17th century, BCE), 皇历 (Imperial Calendar), and 旧历 (Former Calendar), as its name suggests, is designed for agriculture, and of course follows the sun. But not any sun. It’s the sun as observed in 中原 or the Central Plains of China, which refers to the entire region along the Yellow River, and to 河南 (Henan) or “south of the Yellow River” in the narrow sense. The 二十四节气 (24 solar terms) are set according to the sun. But each month is set by the moon (of course!) and in an agricultural calendar year months are often added to align with the solar year. The month is lunar, and the year solar. This is roughly how an Agricultural Calendar Year operates. And this lunisolar calendar is designed by and for the farmers of the Central Plains, not other Chinese regions, and of course not countries like Japan, Vietnam, and Korea whose climates are very different from that of Henan.
The Spring Festival is to celebrate both the beginning of spring in China, or precisely, the Central Plains of the country, and the beginning of a new year of the Chinese calendar. And even if the festivities, whatever name they might fall under, are to celebrate the coming of a new year, it’s a Chinese calendar year. After all, the Agricultural Calendar mentioned so often above is often known in English simply as the Chinese Calendar.
In short, if anyone are having festivities, they are celebrating a Chinese festival or the beginning of a Chinese calendar year.
People in Beijing and Heilongjiang, where it’s still winter, or in Hainan, where it’s tropical in the southernmost part of the island, or hotly tropical far away on the outlying islands like Sansha City in South China Sea, celebrate the New Year and the Spring Festival, because they are Chinese regions. The neighboring modern-day countries in East Asia that is also known as Sinosphere celebrate it because, for longer or shorter, more recent or distant historical periods of time, they were Chinese or believed they were more or less Chinese. And the celebration has been inextricably built in their customs and culture over a course of hundreds of years. And their emigrants to Western countries and the rest of the world, and their descendants celebrate it because their ancestral countries celebrate it. Others celebrate it because they like holidays.
And nobody should get offended when hearing “Happy Chinese New Year”, if they insist that they are celebrating Lunar New Year, instead of Chinese New Year. This is because the wish is, as shown by the term “Chinese” in the greeting, intended for those who do celebrate Chinese New Year. Or put in another way, they are greeting someone else.
But Chinese people are just equally happy when people wish us “Happy Lunar New Year” because, as noted above, we do informally refer to the calendar as a “lunar” calendar, and the year as “lunar” new year. It’s only that the calendar is agricultural and lunisolar (lunar month, but solar year) and that Chinese people sometimes love being scientific. The Chinese calendar is about being agriculturally scientific.
But language isn’t about being scientific. It’s about being idiomatically, socially, and politically correct, but not necessarily scientifically so.
So, this might be why it’s a problem calling it Chinese New Year and many people prefer to be greeted “Happy Lunar New Year” in some countries, particularly where being Chinese carries a sociopolitical stigma. They want to dissociate themselves from China, are ashamed of being Chinese, or don’t like being mistaken for Chinese.
But those who insist what they are celebrating is Lunar New Year, not Chinese New Year, should understand that they wouldn’t have had a festival called Lunar New Year to celebrate historically in their ancestral countries and that nobody would have been bothering to wish them “Happy Lunar New Year” today in other countries if not for China. This is because, as explained above, it was China that invented the day and the calendar in the first place and brought the tradition to their ancestral countries; and because it has been precisely the ever rising influence of China that brings the festival to the greater attention of the world.