As you may remember, I said that I have a Bachelor of Science in "Natural History and Interpretation" ( http://blogevolved.blogspot.com/2013/03/introducing-hadiazmy-1st-listmania-list.html ). Thus, I had a lot to say in my comment on Ben's "Phylogenetics is Moon Man Talk" ( https://extinctmonsters.net/2016/12/21/phylogenetics-is-moon-man-talk/ ). This post is a modified version of said comment. Here's hoping you get as much out of Ben's blog as I did. It's been very influential to my reviewing ( https://www.amazon.com/review/R1Y51RJP1YORCC/ref=cm_cr_dp_title?ie=UTF8&ASIN=1416938575&channel=detail-glance&nodeID=283155&store=books ).
P.S. Happy New Year!
Quoting Ben: "People are introduced to these categories in grade school, and you’d be hard-pressed to find somebody who couldn’t tell you whether (say) a cat is a mammal or a reptile. What is missing is what that actually means. We can’t assume that just because somebody knows a cat is a mammal, they know that fur and milk glands…are things to look for when categorizing mammals. They also may not know that “mammal” is an evolutionary group – that all the animals that fall under this banner are more closely related to each other than they are to anything else."
If that's the case, then I'm surprised, given that even I (I.e. A little dum-dum who grew up in various small hick towns) knew & heard all that in grade school: That some animals are more closely related than others; That the more closely related ones share certain features that others lack (E.g. In reference to mammals, even whales have some body hair). My lifelong interest in dinos & educational tv might've helped, but I still didn't know much else about evolution until college. Speaking of educational tv, the "Who's Who?" episode of "Kratts' Creatures" may be the best children's tv explanation of how animals evolved (& thus, should be required viewing for anyone who talks to laypeople about phylogeny).
Paraphrasing Ben: "as Torrens and Barahona demonstrate, [cladograms] are regularly misinterpreted by the public."
Cladograms on their own, yes, but if an educator is using one like you describe ("How can educators hope to cover so much ground without confusing, distracting, or alienating their audiences? One option is to use a cladogram, or evolutionary tree"), then that shouldn't be a problem b/c the educator is there to clarify the cladogram.
Quoting Ben: "Basic Vertebrate Classification…Evolutionary History Through Deep Time"
You're obviously much more intelligent/experienced than I could ever hope to be. However, I feel like maybe I can provide a different perspective (& thus, a possible solution), given my personal experience as a little dum-dum who had to figure out a lot of that on his own through trial & error.
When laypeople ask me what something is, I 1st ask them if they know what reptiles/mammals/etc are & then describe the something accordingly. For instance, when talking about dinos, I describe reptiles as "4-legged backboned animals characterized by keratin scales (among other things)", dinos as "land-living reptiles with an erect posture", birds as "flying (or secondarily flightless) feathered dinos", etc. In your case, I'd describe mammals as "4-legged backboned animals characterized by body hair & milk glands (among other things)" & non-mammal synapsids as "proto-mammals, or extinct relatives of true mammals".* Also, when asked why something (E.g. Pterosaurs) isn't part of a certain group (E.g. Dinos), I say, "All dinos (including birds) share a common ancestor & certain features inherited from that ancestor (E.g. An open hip socket). Pterosaurs lack said features, which is how we know they're not dinos." If/when need be, I explain that said features might seem small/insignificant to us, but make a big difference in the evolution of said animals (E.g. An open hip socket allowed the erect posture of dinos, which allowed them to run faster & grow larger than other reptiles). Does that help?
*Seriously, educators should use "proto-" more often. It helps a lot when describing intermediate groups to laypeople (E.g. Non-dino dinosauromorphs = proto-dinos, or extinct relatives of true dinos; Non-croc pseudosuchians = proto-crocs, or extinct relatives of true crocs; etc).
Quoting Ben: "How Scientists Discover Evolutionary Relationships"
Have you read Hedley's "Dinosaurs and Their Living Relatives"? If not, I definitely recommend doing so. It may be the best children's dino book when it comes to explaining that (& thus, should be required reading for anyone who talks to laypeople about phylogeny). I like it so much that I reviewed it in "Cladistics yay!" ( https://www.amazon.com/review/R1SCM65CLPZD4M/ref=pe_1098610_137716200_cm_rv_eml_rv0_rv ). Here's hoping you like "Cladistics yay!" (&, assuming you have an Amazon account, vote Yes for it ;) ).