You explained well. They also used to count age in the same way in Japan. People were 1 as soon as born, and everyone went up another year on new years day, so someone born in the 12th month would be born 1 year old and then shortly be 2 years old on new years. A little strange, so I can see why they switched to the Western way. No one uses the old system anymore in Japan, but I think most people are still aware of it. The Japanese use the Chinese pronunciation for 立春 but they change it a little to make it easier for them to say, making it risshun (ri shun).

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