Your argument assumes that the recruiter simply asked for a Word doc, using it as synecdoche (intentionally or not) for a formatted text document of unspecified type and doesn't really care what format it's in as long as she can read it. But this is demonstrably untrue, because I provided a PDF before she specified a format, she was able to read it, and then specifically asked for a Word doc.
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Bitt Faulk