I agree that the differences are geographic and linguistic more than culinary, but it's always seemed to me that sauce for a chili dog more likely will contain beans and/or tomatoes where as Coney sauce rarely does. The sauce on a Texas hot wiener is more like the latter. Am I right, John?
The sauce on a Texas Hot Wiener, or Coney IS more like the latter picture. I've been to many hot dog establishments in the Northeast region of the country and in my experience most chili dogs do not contain beans. There is no set rule. In New Jersey where I live the great majority of places do not have beans in the chili. But I've come across many places that serve chili with beans.
It gets sort of complicated. Here there are trucks and carts that serve Sabrett beef hot dogs. Many of them have chili that do not fit the definition of Texas Weiner or Coney. The chili can be canned versions like Castleberry and are tomato based. Many vendors use a packaged chili (with or without beans) and doctor it up. Many places like Father & Son and JJ's truck serve a hot chili (spicy) similar to the Kuhn's chili that is popular in Connecticut.
When we talk about Texas Weiners in Jersey, there are 2 basic types. The Paterson area "Hot Texas Weiners" and the Plainfield area Texas Weiners. Paterson (including Clifton and all of Passaic County) Texas Weiners are deep fried pork and beef franks that are served with mustard, onions, and a thin chili sauce that is heavy on spices such as cinnamon, nutmeg, allspice, etc. The brand of dog is often Thumann's made for deep frying. Places that fit into this category are Libby's, the Hot Grill, Pappy's, Johnny & Hanges, and Falls View. There are many others.
The Plainfield area Texas Weiner is a grilled rather than deep fried pork and beef frank that is topped with a thicker chili. It has some of the same spices that the Paterson area Texas Weiner does, but it is hotter/spicier. Almost all of these places use a Grote & Weigel griddle frank. This type of Texas Weiner is close to what many call a Coney. Examples of this style include Texas Weiner l and ll in Plainfield and Greenbrook, the Red Towers, J&G in Dunellen, Manny's Texas Weiners.
The Coneys I've come across in the Hudson Valley New York, PA, New England, and even some places in Jersey include a thicker chili sauce (again beanless) similar to the Plainfield area Texas Weiner described above; but it isn't as hot/spicy. Similar spices such as nutmeg, allspice, etc. Most of these places are either owned by or were started by Greeks. The Coneys I've come across are mostly beef/pork dogs, but the brand varies. While Jersey Texas Weiners for the most part stick to Thumann's deep fryer for one style and Grote & Weigel for the other (both high quality dogs with casing), the Coney joints use a variety of brands and either skinless or natural casing. I've come across Hatfield, Berks, and Kunzler in PA, and Tobins, Hatfield, Grote & Weigel, Kayem, and other brands in New York and New England. There is also more variation in the way the dogs are prepared. They can be grilled, fried, or steamed.
In short, all dogs with chili are chili dogs. Not all chili dogs are Texas Weiners or Coneys. These have subtle differences depending on region, but for the most part use a beanless chili, have a spicing that has Greek origins, and use a milder beef/pork frank. A dog labelled a chili dog can be any type or brand with chili that may or may not have beans, can be tomato based or not, and can be extremely spicy. They can be prepared any way.