Hi,
As an addendem to Brian, due to the limited shelf life of milk (not just cream), the milk train had the highest priority (at least on the Pennsy where my information comes from former PRR workers in the family) of all trains. Every other was to get out of the way or stop. The fastest locos were assigned to lighter consists.
Since there was not to be a worry about opposing traffic (a very easy run), the term "milk run" became the easiest, least worrisome (fewest decisions needed to be made) task good for an newcomwer to an orginisation (if in the military - to give a person experience at the next higher level for the least risk).
On branches and short lines where the expense of separate milk trains were not justified, these would be the passenger trains (or mixed trains) that operated on the line.
FWIW and IMNHICR (I might not have it completely right).
Doug vV