If you're big into dictionary definitions, wiktionary has the following:

"A star, especially when seen as the centre of any single solar system"

And OED lists: "any star around which planets move"

The thing is, the usage of a word determines its meaning, not any one definition. And the word sun is in fact habitually used to mean 'star,' just read or listen to works of sci-fi I guess. If you pretend it's not, you're simply not capturing the full meaning.