Final answer:
Option C: '...Proteus repeated to the Duke of Milan what Valentine had told him.' is the detail that refines the theme of betrayal, showcasing Proteus's severe breach of trust towards Valentine.
Step-by-step explanation:
The detail from the text 'Who is Silvia' from the Shakespeare Story-Book that refines the theme of betrayal in the passage is option C: '...Proteus repeated to the Duke of Milan what Valentine had told him.' This act of sharing a secret not meant to be divulged demonstrates betrayal, as it discloses Valentine’s trust in Proteus and Proteus’s subsequent breach of that trust. In comparison, options A and B describe a developing relationship and trust, while option D narrates the betrayal without providing a specific example of it.