I agree with your last paragraph (see my last paragraph). But I think the most natural reading of the initial task description is that the machine also pretends to be a woman. The line “we do not wish to penalize a machine for being unable to shine in beauty competitions” supports this interpretation, given that a beauty competition is an event for women, under the assumptions of the time. So I think there are conflicting cues in the paper as to the intended interpretation.
As you say in your other comment, though, I don’t think Turing thought the exact details of the game were important - which explains why he didn’t trouble to spell them out very exactly.
If I had to guess, I’d say that Turing assumes that as the machine has no gender, the only relevant difference between the machine and the woman is that one is human and one is not. So for the rest of the paper he focuses on that difference and is vague on the gendered aspect of the task.
