I have always thought of it as an old term that includes Birkenhead and the wirral side along with Liverpool and it's northern and southern suburbs.

Even if you just describe the docks, "Liverpool" would not include the ones on the south side of the river.

The term "Liverpool" is pretty restrictive in describing the area - Merseyside is much more encompassing and descriptive of the populated/inductrial area - even before London used it to make a new county like area...