Working part time while studying in USA on student visa. F1 visa holders cannot accept off-campus employment at anytime during the first year of their studies. Under certain circumstances, the U.S. Citizenship & Immigration Services (USCIS) may grant permission to accept off-campus employment after one year of study.
Hereof, are you legally authorized to work in the United States?
You are confusing legally eligible with legally permitted. The question "Are you legally authorized to work in the United States?" implies you are already in possession of a visa valid for work, are a US permanent resident or a US citizen.
What does it mean to be authorized to work in the United States?
If an individual is not a citizen or a lawful permanent resident of the United States they will need a permit to work, officially known as an Employment Authorization Document (EAD), to prove eligibility to work in the U.S. It is the responsibility of both parties to show and require proof of legal employment status.
Are you legally authorized to work in the United States without sponsorship?
It means "Are you a citizen or legal resident of the United States, or do you hold a visa that allows you to work in the United States without being sponsored by an employer?" It means "We are not going to sponsor you to work in the US.