Insurers take an interest in all of the changes you make to a car. Things like taking off badges and adding spoilers seem very innocent because they don't make the car go faster, so why tell them, right?
But this simply isn't how they work. It's not just about the actual performance of the car. In fact that is secondary. What really interest them is the type of person driving the car and the how big the risk of them crashing it is. And the kind of guys who modify cars are more likely to crash them!
Yeah, bleat about it all you want but it's a fact. Why else would insurance companies refuse to insure modified cars or charge more for them?
So they win, whatever you do. If you don't tell them about the mods and don't crash then they've got your money and not had to pay out. If you do crash and they find out about the mods, which they will, they've got your money and...they still don't have to pay out!
As long as your tyres meet the specification for the car, or are higher, you don't need to tell them. Badges, I'd be surprised if they bothered much about that on its own but if you'd done a rake of cosmetic mods then yeah, I would definitely tell them.