The trouble with this is that you're taking a mathematical idea ('divide by') and wrenching it out of
context. It's been done before: we use the words "
orthogonal" and "
modulo", for example - both mathematical in origin - quite happily, outside of purely mathematical contexts, but that's because they've been given a
meaning that is useful in the places where they're used. My argument will be that the new "linguistic" (contrast: mathematical) usage proposed has no such useful application, and is therefore nonsense.
Now, if you take your eight apples and 'divide' it into 'no groups', resulting in no apples, then where do the original eight apples go? I challenge you to do this with eight real apples, and make them all disappear by 'dividing' into 'no groups' (eating them isn't allowed - that's dividing them 'into' one person. :-)
Language (including mathematical language) just doesn't work like this - it's as though I were to say "I'll multiply the moon by a cow, and look - it gives me forty stars! Prove me wrong!" Unless you can provide some stable context (in this case - apples or moon - it needs a corresponding physical reality, because the mathematical one plainly doesn't apply) then you're just talking nonsense, and there's no reason to take you seriously.
Maths (complicated maths, for which there's no obvious physical analogue) gets away with it, because the system of rules creates a consistent and stable context which determines a semantics for the mathematical statements - which is why mathematicians can talk about such things as the square root of minus one, and still make sense. The sense is provided by the consistent universe of discourse (consisting of mathematical objects) to which the statements about sqrt(-1) refer (i.e. objects in the complex plane, clifford algebras, etc.)
If you make up new language without having such a stable context, there's no reason to expect anyone to understand or believe you, because there's no way to check the application of the new term, and there's no way to use your language to do anything useful. New language 'sticks' because it turns out to be useful in dealing with something, whether it's another part of the language (like in maths) or some regularity in our observations of the world (like in science) (or, going out on a limb, because it strikes a chord 'within', perhaps, as in poetry.)
With the best will in the world, I don't think your extension of the term 'divide by zero' accomplishes any of these!