You have a point on the mother/daughter thing. Sometimes, the jargon break down is mother sauces, daughter sauces and little sauces. However, modernly and "technically" all of the sauces which are not mothers are referred to interchangeably as compound sauces, and I believe this was Escoffier's terminology as well. However, while I can read a recipe in French to the point of being able to follow it, I don't actually read French, so can't be sure what Escoffier's terminology was.
You have to remember, cooking isn't physics. There's no universally agreed upon terminology. Furthermore, the migration of French terms into English is extremely uncertain. English gets in the way and adds a number of words on its own which may or may not have exact French parallels. There's a thread from last month that concerned the difference between "stock," "broth," and "soup." All of us who worked in "French" or "Continental" or other "Fine Dining" kitchens had the same feeling as to the distinctions -- but there's no difference in French.
Also, I did not refer to demi as a "mother," but as an intermediary on the way to other daughters.
There are not, nor were there ever, seven "Mother" sauces at one time. A good mnemonic (well, my mnemonic) for Escoffier's mothers is, "they are the Very BETH." That is, VBETH. Veloute, Bechamel, Espagnole, Tomate and Hollandaise. Allemande had dropped from favor before Escoffier wrote the Guide -- but it's interesting (I guess) to note that he almost single handedly brought it back when he was in France for the Great War, by creating a couple of Allemande daughter presentations. IIRC, the big one was some sort of fish Parisienne. But war over, and it was back to London and Allemande wasn't around enough to be considered a mother.
You understand the relationship between Careme and Escoffier somewhat differently than I do. You see the "student and disciple" line around the internet a lot, but it's a bogus.
Careme cooked and wrote about 100 years before Escoffier. Sometimes it's said that Escoffier assumed the "mantle," but their approaches and styles were very different. It's fair to say that Escoffier was no more of a student or a disciple of Careme than he had to be. To the extent that he was a student, Escoffier was certainly a student of cooking and Careme was the great, dead authority. But, Escoffier's tendencies were to simplify and lighten the "classics," and to make them distinctly his own. He was more revolutionary than disciple.
It's important to remember that although he wrote the Guide, which was certainly the most important cookbook in France of its time, Escoffier wasn't exactly the be all and end all of actual French cooking. Not to diminish his importance or his greatness, but he was primarily based in England and his actual cooking was not that influential in France.
Just as important to modern French cooking are Pellaprat and Mme E. Saint-Ange. You can't really disassociate them from Escoffier, but you can't diminish their contributions to "also starring," either. In fact, if you're interested in the "classic" cooking of pre World War II France, you'd be much better of with a cookbook by Pellaprat or Saint-Ange than with the Guide. I can tell you with confidence born of being there that these were the two sources from which California Cuisine sprung, and believe also that Nouvelle Cuisine and Cuisine Gourmand are based upon them as well. In other words -- the entire edifice of modern "French" cooking.
I think it's a mistake for home cooks to overemphasize Escoffier or to take too much from his cookbooks. It's fun to connect with the history of haute cuisine, and fun to sample the tastes of the well-off dead, but it's not what cooking is anymore, and not a particularly good way to get to it. Escoffier's greatest strengths as a chef were his willingness to modernize and adapt while mainataining the highest standards. For instance, he developed the (then) modern kitchen brigade. As a food scholar and writer his greatness lay in the way he organized dishes (haute, impromptu, regional, bourgeois) and his openness to a variety of cuisines.