Well, the original book, which just had example sentences, was Mimi Kara Oboeru N3. I checked in Routledge's Japanese: A Comprehensive Grammar to try and find more information, but couldn't find a satisfactory explanation. This book didn't have the meanings for ~ように broken down in the same way that the link that Toritoribe shared does, and certainly, seeing it laid out in that way helped me to understand it. I'll be using that site to look things up more often.
Seeing it from that point of view, I think my lack of understanding of ~ように prevented me from being able to ask the correct question in the first place.