Maybe its worth explaining how alternators works, and then other things make more sense.

The Field is a spinning electromagnet. The magnet is made by winding wire around and putting electricity into it. The more electricity into the electromagnet, the stronger the magnetic Field.

So the field circuit is the path to get the electricity through this spinning electromagnetic that creates the magnetic field.
A positive wire connects to a carbon brush that rubs against a copper ring on the rotor. This ring then connects to the windings. The other end of the wire windings are connected to another ring, a brush and then a ground wire.

Power through the field can be regulated by controlling the positive wire, or the ground wire. When controlling the positive, the negative side can be grounded to the metal case. When controlling the negative, the ground wire must be insulated from the case.

Regulators that control the negative (ground) still need a positive wire connection. The positive informs the regulator of system voltage.