It's not ambiguous at all.
ALL Permanent Residents are required to carry their "Green Card."
All Nonresident aliens are required to carry their Passport with a valid US Visa.
Green Card holders are Told about that requirement when the document is issued.
Visa holders are told about the requirement when they enter the US.
But you forgot one category...ALL citizens of the US must carry "what?" to prove they're a citizen?