It seems bizarre to me to have a policy called "White Australia" (or was it not officially called that?), and then enforce it in such a coy, euphemistic way.
If they didn't want non-white immigrants, why didn't they just ban non-white immigrants?
Is it because Australia was part of the British Empire at the time, and didn't have full freedom of action?
Enforcing the policy through a subjective test allowed it to be expanded to exclude politically undesirable people.
The Immigration Restriction Act (1901) required immigrants to be able to "write out at dictation and sign in the presence of the officer, a passage of 50 words in length in a European language directed by the officer". Since the language could be chosen by the officer anyone who couldn't speak all European languages could be excluded.
The most notably example of this is Ego Erwin Kisch who was a multilingual Czech communist. He passed the test in a number of languages until immigration officials managed to find an officer who could speak Sottish Gaelic. Kisch then failed and was convicted of being an illegal immigrant. Although that was overturned on appeal.
Source: McNamara, T. (2009) "The spectre of the dictation test: language testing for immigration and citizenship in Australia". In Extra, Guus, Massimiliano Spotti, and Piet Van Avermaet, eds. Language testing, migration and citizenship: Cross-national perspectives on integration regimes. London: Continuum: 224-241.