Why DNA tests for Indigenous heritage mean different things in Australia and the US

Last week, Massachusetts Senator Elizabeth Warren released a video strongly suggesting two things: she is running for US president in 2020, and she has Native American ancestry.
The second claim was apparently confirmed by the results of a DNA test, which compared genetic data from Warren’s whole genome with that from people of known Central and South American ancestry.
Warren has come under fire from both sides of US politics for releasing her genomic information. Many have questioned the veracity of the test. Others have said that even if Warren does have Native American ancestry, that doesn’t make her Native American.