That sounds right: When I exported my cat from Taiwan to Canada in 2007, I got the paperwork I needed from Taiwan. The paperwork was issued by the Bureau of Animal and Plant Health Inspection and Quarantine, Executive Yuan, Hsinchu branch. Canada, like the US, is a rabies-infected country, so the health re-requisite for importing was up-to-date rabies vaccination. Canada accepted the standard health certificate form issued by the Government of Taiwan (we did not import in the US so I can't speak for US customs).
When I exported my cat from Taiwan to Canada in 2015, I got the paperwork for import to Taiwan from a Canadian Government website. There were many prerequisites for the importation and the forms were specific to Taiwan. In that case, the health certificate was filled out during an examination by my vet, and stamped at a Canadian Government office. We imported/exported in Korea on the way, and the paperwork procedure was similar.
You could find out from the plant and animal inspection office in the US if they require a US-specific health form.