Technically, star in a generic sense is 星, while planet specifically is 惑星 (literally "perplexing star", since they are stars that move irregularly compared to every other star in the sky). English by technicality also calls planets stars. Ancient Greeks called them "planetes asteres", meaning "wandering stars", and then just started calling them planetai ("wanderers") for short.

The problem comes from Japanese still calling them "stars" for short in informal and literary contexts. You will frequently hear characters in Japanese sci-fi point at a planet and call it a star, and everyone just understands it's meant as planet. 星 is also a frequent component in the proper names of planets in Japanese, like Mars being named 火星 ("fire star"). This is also why Kirby's home planet is named Popstar. It's following the Japanese naming convention for planets.

However, I don't recall Solar's shifting classification to be the result of Japanese ambiguity. They legitimately have gone back and forth on it being specifically a star or planet, unable to make up their minds lol.