Asking for definitions is not allowed here, so I will give an answer based on my personal understanding of these words without consulting a dictionary, which you can of course do anyway.
Rocks and stones can largely be interchanged in everyday language. You are incorrect to think you cannot call a piece of mineral matter from the Earth a stone. In fact, if anything, the word 'stone' suggests a clump of mineral of smaller dimensions than the word 'rock'. If the piece of mineral matter were large enough to sit on, for instance, I would be unlikely to refer to it as a stone, but instead, a rock.
The word 'rock' also implies to me more irregular dimensions than the word 'stone', which for some reason draws images to me of a smooth, curved surface, while a rock seems irregular and rough.
These are just my conceptions of these words, and I'm sure others will differ. But a hard-and-fast rule would probably be that they are interchangeable.
(The term 'stone' is also used to denote a building material, whereas 'rock' is generally not used in this context I think.)