If milking by hand, each cow would normally have a pail or bucket put underneath her.
From Wikipedia (emphasis mine):
The first milking machines were an extension of the traditional milking pail. The early milker device fit on top of a regular milk pail and sat on the floor under the cow. Following each cow being milked, the bucket would be dumped into a holding tank. These were introduced in the early 20th century.
Note that the author of the definition used there couldn't even make up their own mind between pail and bucket. (Or perhaps different authors were responsible for different parts of that entry.)