The web plugin actually falls back to 'web page with snapshot' if it can't detect a journal paper, which in many ways is better than trying to drag around big PDF binaries.

Thanks! I've used zotero only minimally, mostly to support users of my software who use zotero, rather than using it myself for my own purposes.

Is the "web plugin" something built into zotero, or an extra plugin to install? Calling it "plugin" makes it sound like something extra to install, but not finding it easily googling. Help me out?

I think OP talks about the Zotero Connector Browser Plugin. So its not a plugin for zotero but your browser (Firefox, Chrome, Safari and Edge are supported) The plugin ads a button to the Browser where you can add the currently viewed page to your database. It even tries to detect references to papers or other content and choose the matching entry type. Eg on Google Scholar you can choose which search result you want to add. It its a PDF the file gets saved to your zotero database.

All in all very handy.

Does it also save the JavaScript and images, so that the page can be "archived" locally?

You can link to any type of file and it will open in the default program for that filetype when you double click