Distinction between "who" and "that"

It has been said that the obligatory use of "who" in reference to people is a myth, yet I understand that there are times when "who" is obligatory and times when "that" is quite acceptable, if not required. How does one differentiate between the two? Is "who" obligatory
only when people are uniquely named
("John, who ..." but "The John that ..." and "A/The student that ..."),
when a specific person or specific persons are under discussion
("Your English professor, who ..." and "My father, who ...", but "A/The truck driver that ...") or
when the subordinate clause does not identify the person concerned
("The English professor, who is always considerate" implies only one English professor, who happens to be considerate, but "The English professor that is always considerate" implies more than one English professor and refers to the considerate one)?