I think we agree on the facts, I just found your explanation a bit unclear/misleading. And the site you linked to didn’t label the dots, so it was hard to see what is really happening.
It will if you tilt it 
Despite the left turn the thread took, I appreciate your answer. It did explain exactly what I was wondering about. Thank you.
About that “vertical gnomon”, as in you just stick a rod pointing straight up somewhere and tell time without a node, just using the direction of the sun, the short answer is that not only does setting that up still depend on the latitude, your hour indicators will not be straight lines and will resemble something like
Basically the solar azimuth depends on your latitude as well as the declination and local solar time. Now those horizontal garden dials are not constructed with a vertical gnomon, however the horizontal orientation of the face skews the hour lines depending on the latitude as you can see in that link above, or download that free Shadows app and mess with the latitude setting to see for yourself.