I believe in the old days there were levels of student. Often, there would be a general class, which served as a probationary sifter of sorts. Only one or two basic forms were taught, lots of endurance type activities, and only the most coarse of applications and details provided. Often the sifu himself didn’t teach this class, but seniors did.
From this group, some students would be bai si’d (bow to teacher, usually a tea ceremony where the student is formally accepted into the school) and they would move into a more select group where intermediary forms were taught, more details, more application, and closer bond would form between student and teacher, who would often teach more personally as well.
And from this group, a few individuals would be chosen over the years for Yup Sut Dai Jee (Inner Room Student). This would often involve another ceremony, sometimes with the parents giving money to support the student to actually live in the room with the teacher, and the bond formed would be akin to father and son, where the student could even enter the sifu’s bedchamber (inner room) without pause. These students learned the advanced sets, details, and applications, and also how to teach the art to others and spread it.
Sometimes, especially in the later years, a sifu might pick one of the Yup Sut Dai Jee to succeed him/her, to become the Cheung Moon Yan (Gatekeeper) of his teachings. Though it could have been, it didn’t have to be the best fighter or most talented student (those, especially if taken early on, would sometimes have passed away or even striked out on their own or moved to a different area), but it was someone who could remember (or at least read from records) all the movements of the art and be trusted to preserve it for the next generation, especially as time passed and questions arose.
Usually, I believe, in the general class, you studied with X sifu, as there was still some degree of separation. When you were formally accepted, you studied with sifu. At that point, there was no debate over the relationship.
(It should also be noted that Sigung, Sihing, Sijay, Sidai, Sibak, etc. are all teacher/relations and are covered by that, meaning you can study with them anyway without needing a more formal sito relationship, though in some cases if it becomes primary, it can be formalized that way as well).
RR