Academia in general sticks surprisingly close to the guild model. You have professors as masters, postdocs as journeymen, grad students as apprentices, and undergraduates as initiates.
A professor then is simply a master who, rather than earning a living as a craftsman, has a position and salary paid by a patron or institution. One might compare to a painter who became a salaried court painter, or one who received regular stipends from the Académie de peinture et de sculpture, rather than earning an income from seeking commissions or giving lessons.
I think the common way people view the situation has changed in part because these craft occupations are more difficult to sustain, so you're not really seen as "done" until you've landed a regular full-time job. Like being a painter without a salary, being a lecturer without a salary is nowadays a very precarious position. There used to be enough wealthy patrons paying enough in portrait commissions and tutoring fees for such careers to be make for a decent middle-class living, but it's not a common career anymore, so now people typically either look for a full-time academic position, or look for a full-time job elsewhere. You can temporarily hang around academia as an adjunct lecturer, but it's not a good or stable living.
Closer, but still not quite right. The concept of graduate students is quite a recent one; and until recently the MA was the standard undergraduate degree.
This is still seen at Oxford and Cambridge: After 3 years of study, you write examinations and receive the BA, whereupon you leave the university; 4 years later, you receive the MA. (And until very recently, upon applying to receive the MA you had to certify that you had "continued to practice" in the intervening years.)
Unlike in the U.S., Masters and PhD students are therefore not lumped together as "graduate students". Bachelors and Masters students are instead lumped together as "students", i.e. people who are taking an undergraduate education, often living in dorms, and receiving a small stipend (the SU) to support their cost of living. PhD students, by contrast, are junior research staff, employed with a proper salary (typically ~$50k), living in their own apartments or houses, entitled to attend departmental meetings, etc.