How are magnetic fields 'detected'?

While reading Griffith’s Electrodynamics, I didn’t quite understand how magnetic fields were introduced.

Electric Fields were introduced as an intermediary calculation to finding the electrostatic force, and their detection makes sense in a way, because the Field and Force are in the same direction. Specifically, the electric field (a physical entity of its own right) is the force per unit charge.

However, magnetic fields were introduced as ‘that which is detected by bar magnets’. We used the bar magnet to infer that the magnetic field outside a straight current carrying wire makes concentric circles, and then brought another current carrying wire to see the direction of the ‘force’. We then inferred that the force law must be of the form $q(v times B)$.

My question is this: How did we conclude that the bar magnet detects the field and not the force, and how does it detect the field and not the force? What is the difference between what happens in a current carrying wire and a bar magnet in the presence of a magnetic field?