You are correct that the Torah prohibition refers only to meat and milk that were cooked together in a way that their tastes intermingled.[1] However, the Rabbis extended the prohibition to not eating meat and milk together even if they were prepared separately, and gave guidelines as to what is considered eating together. Following the eating of milk products, one needs to take steps to ensure that his hands and mouth are free of dairy remnants before eating meat.[2] Following eating meat products, one needs to wait before eating milk products. Two reasons are given: 1. Pieces of meat are likely to remain between the teeth, and the situation resembles eating meat and milk together.[3] 2. The taste of the meat remains in the digestive system a significant amount of time.[4] There are several halachic differences between these two reasons, with practical ramifications.
The gemara[5] tells of Amora’im who, after eating meat at a meal, would not eat milk products until the next meal. Since the reason for waiting has to do with actual physical conditions, the Rishonim understood that “the next meal” must refer to a set amount of time between the last eating of meat at one meal and the eating of dairy at the next one. Several opinions and customs arose in applying the concept. The most prevalent ones relate to the Rambam’s[6] statement that it is “like six hours,” which gave rise to opinions of a full six hours, just over five and a half hours, or a bit more than five hours. A minority view gives a minimal break of one hour between meals (Dutch Jews follow this custom today)[7], while a third approach compromises with three hours (German Jews). All should follow their family minhag.
Whereas some require waiting only after eating actual meat (not meat gravy alone), most wait six hours after eating a food cooked with meat.[8] However, after pareve food that was cooked in a fleishig pot, one need not wait at all,[9] even if there may have been a bit of residual gravy in the pot.[10]