For 3, I imagine clarifications in parentheses are okay, maybe good enough, for the math domain. A more high-powered solution probably wouldn't come in handy all that often, and tables would probably help with the visual clutter. But I wonder as Arbital turns its eye towards more subjective domains whether the problem of describing theories or taxonomies that vary somewhat in the wild will come up more frequently, where statically explaining the multiple conventions in use ends up being considerably more complex to read than giving a more dynamic presentation. Less "this is a feature you need", more "this is an interesting opportunity to deliver a better learning experience than a textbook or encyclopedia article could provide".

Here are some things I'd love to be able to add to this page to make it more explanatory (some of these might be technical features, some an invitation to brainstorm more creative visualizations):

• Tables! Specifically, a table of the structures going down and the properties they require going across would be a much clearer way to summarize this information. I tried HTML tables and preformatted-text ASCII tables here but couldn't get either to work.
• Some indication of what collection of properties imply other properties, or what collection of properties don't get their own name because they're logically impossible.
• A way to toggle between different naming conventions, as described in the text when different authors assume different axioms with the same word.
• Visual highlighting for the most important members of the tree—if you're just starting out in abstract algebra, knowing what a field is is much more important than knowing what a rig is.