Welcome to using WebMention, Jeremy. Still figuring out how to best use it myself with regard to how they get displayed on my site.
Like you I use WordPress, and I would love for mentions to display more like the old pingbacks, where you’d get a snippet from the mentioning site from around where it links to you. Now it mostly is ‘site x mentioned this.’ which makes me click to get a notion if it’s relevant.
On Webmention tweaks I documented some of the things I tried. The issue is that because the tweaks are in the Semantic Linkbacks plugin, not in the WP theme, you can only make those tweaks a permanent option if it gets rolled into the plugin (no such things as a child-plugin like with themes). And I’m not confident enough of my changes to figure out and try submitting them to the maintainers of the plugin.