Not an expert with the ticket dispatching rules, but I do have some that work for me.
Based on what I see in your screenshots it looks like both the Library Tickets and the BYOD Tickets have the same email from address. I suspect that's where the duplication is coming from.
I would change the rule conditions from 'To E-mail Equal libraryrequests...' to 'Subject Like Library Tickets' and create another rule for 'Subject Like BYOD Tickets' and see how those apply to incoming emails.
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