Quote from Churchill Policy Book
"Changes made to your vehicle which improve its value,performance,handling or attractiveness to thieves."
That sounds sensible to me (although different insurers may have different definitions, as Dave1 has illustrated). In Tango's example, I would have thought that installing a non-OEM exhaust rear box which is just a bog-standard one from Kwik-fit or similar would be no issue. But installing an HKS back box probably would be an issue, since it could be argued that it improved performance (albeit just slightly) and also the car's appearance. Similarly LED/HID lights should be declared, I would think.
Installing a different make of tyre wouldn't really seem to be an issue to me unless it was a particularly expensive/exotic tyre, but installing bigger alloys than the OEM ones would be a modification.
I would say, if in doubt - declare it !