How to handle clause claiming developments "relating to the business of the company" in a large company?

In practical terms, what this boils down to is that you offer your inventions to your employer first. If they claim these, you will generally be rewarded to some degree, possibly indirectly; details depend on the invention and the company. If they tell you it's outside their scope of interest, that frees you to do whatever you wish with the concept; you can and should get that in writing.

Yes, it's fuzzy. But the argument is that part of what they are paying you a salary for is your creativity, including ideas that arise when you are not on the clock.

Good luck finding an employment contract which does not contain this clause, unless you are in a country which explicitly forbids it.

You can ask them before examples of where they draw the lines and how a hypothetical creation would be handled. For example, my employer permitted involvement in open-source projects without needing to examine every idea; they trusted that employees were clever enough to recognize when something was a big enough breakthrough then it should be called to the company's attention first.

It is worth checking with a lawyer. There is no way that you can know if they have some other group developing something that matches a side project that you are working on.

I would assume that it is very unlikely that this would ever become a problem, but that is what lawyers are for.