Is it truly correct to say that the Earth orbits the sun?? Wouldn't it be more truthfull to say that the Earth orbits a common center

Absolutely, and when calculating highly accurate orbits you have make sure the effect is not negligble. But, the combined mass of Mercury, Venus and Earth is probably less than the mass lost by the Sun in a year. Solving the combined effects of 3 planets, the outer planets, suns mass loss etc. is extremely difficult.

I have said this before and I'll say it agin. If when describing something, non-technically, you had to explain every possible option and keep things extremely accurate it would take several books. For the sake of brevity I keep it as simple as I can but hopefully no simpler, not to say I always get it right.

Yelmalio