I doubt that an US based company accepts debit cards from non US banks
Yes, they do, and I can tell you this from personal experience.
But you've got it nailed down. In order for them to accept it, it's not enough for it to be a debit/credit card, you need it to be "compatible" with their system.
If your bank is from Chile (for example) and they give you a debit/credit card that is, for example, Visa or Maestro, you can use it in the USA if the merchant accepts Visa or Maestro.
If your card is DinersClub, but the merchant's system is not compatible with DinersClub, it doesn't matter if it's a debit or credit(or the country) card, you won't be able to use it there.