Active current sensors are simple; the way they work is purely magnetic field strength, not "induction".
Typically they have a magnetic core, that has a gap with a hall-effect sensor in it.
The output from the sensor is amplified and used to drive the feedback coil, which has eg. 1000 turns (vs. usually a single "turn" for the cable being sensed).
The current required to balance the net field back to zero at the sensor will in that case, be 1/1000th the input cable current.
By using a load resistor in series with the feedback coil, appropriate for the current range being measured, you can read a voltage across that proportional to the input current.
(Or you can add another opamp to provide gain & current feedback at the same time, etc.)
Some magnetometers may work in a similar way - eg. how much current in a known coil is needed to null the external field.
Such field strengths are just proportional to current x No. of coil turns; Ampere-Turns.
Proton magnetometers are a completely different animal; in those, a relatively strong bias field is used to magnetically align the atoms in the sensing medium - possibly water!
When the "reset" bias field is cut, the atoms will re-align to the extenal field after some time, depending on the strength of that field.
When they switch back, a small pulse is produced in the bias coil & the time that takes is proportional to the external field strength.
(That's all from memory, so possibly a bit fuzzy - I looked at a DIY design in an electronics magazine back in the 70s?).
Edit - found it! There is a copy of the magazine on the World Radio History site:
The output is a frequency rather than a pulse.