In their answer to What is being depicted on this ancient Greek wine cup (kylix)? /u/Deirdre_Rose stated that satyrs are chimeras of men and horses. I was surprised as for me it was goats (although sometimes with horselike ears) and not horses but they confirmed it was not an unintended mention.
Looking at English and French Wikipedia articles on satyrs, we get mostly the same difference, and then some more with Silenus (for the English Wikipedia, it seems to be only the chief of the satyrs, for the French it is both another name of satyrs later used for their chief). Looking in my 50 years old printed French Encyclopedia Universalis, I get a disappointingly short article, mostly in line with the French Wikipedia but hinting that things are not that clear in the Greek tradition, even more when you add relationship with the Roman one and the identification with fauns. I can't get my hands on my copy of Edith Hamilton's Mythology to see what is there on the subject.
What's the current consensus on satyrs and related beings? How come that there seem to be at least two traditions on them?
I consulted a number of sources in my library to see what they would say on this. The consensus is that there is little consensus. Some sources said that a satyr could be part horse or goat. Some leaned in the direction of asserting that in Greek sources, the horse dominated, while in Roman sources they were more typically part goat.
Any artist or writer - ancient or modern - who asserts that there is a definitive form for a mythical creature is blowing smoke. If a writer or artist insisted that a horse had five legs, we could all call foul; we would be within our rights to demand that this mistaken party go out and simply LOOK at a horse. But when it comes to something like a chimera, the ancient or modern authority must rely on impressions as expressed by the folk. There may be a rigid standard in written sources (although my written sources indicate that there is no such thing), but there would certainly have not been a rigid standard among the folk for an idea spread over centuries and vast parts of the ancient world.
Folklorists find that traditions can vary a great deal in how they are expressed from person to person, village to village, country to country, and century to century. Folklore changes. It is fluid. We try to hang our hats on something to make sense of the jumble, and it seems clear to me that the satyr depicted on the Greek wine cup in question was part horse. So here we have one example of one Greek artist who believed this was the appropriate way to depict a satyr - and this was certainly within proper cultural norms for that time and place. One assumes that the apprentice pot painter who depicted a satyr as part man and part frog had his pot destroyed by the master - and we can assume he was thrown out of the apprentice program. But in the next village, the next century, or certainly the next country (i.e. in the Roman world) featuring a blend of a man and a goat as a satyr was likely to have been more acceptable - perhaps even the norm at some point.
edit: We can hang our hat on rigid definitions of what a satyr is, of course: a satyr IS part horse; this is not part horse; therefore it is NOT a satyr. But we have to keep in mind that a broad swath or European cultures perceived the forests as filled with wild creatures that often took the form of animals or chimeras. We tend to group them together with the term 'satyr' - rightly or wrongly. If we are thinking of these creatures in general, we could find examples of horses and goats. In Scandinavia, the horse is predominant, for example.
One of the things I have noticed with many redditors is that they are influenced by gaming, where rigid forms are enforced - and each form has very specific powers and abilities as defined by the rules. Folklore - and reality - doesn't work that way.