The definitions are standardised so that a contract written in London settles the same way as one written in New York. A determinations committee of dealers and buy-side firms votes on whether an event has occurred, which removes most of the bilateral argument.
Restructuring is the contentious one, because a debt exchange that leaves creditors worse off may or may not qualify depending on which flavour of contract was traded. This is why identical-looking CDS on the same name can be worth different amounts.
Example: a determinations committee rules a failure to pay after a missed coupon and a 30-day grace period. An auction two weeks later sets the recovery at 32, so protection buyers receive 68 per 100 of notional.
Related: credit-default-swap, default, recovery-rate, seniority, covenant