Inductive sensor and equivalent circuit
You have a variable reluctance transducer (tachometer).  

The coil wrapped around an iron core is your sensor, with a reluctance \$R_c\$.  The target is your ferromagnetic toothed wheel, with a reluctance \$R_t\$.  Air gap reluctance \$R_g\$.  The total reluctance of this magnetic circuit is:

$$R_T = R_c + R_g + R_t $$

The magnetic flux \$\phi\$ produced by the magnet is applied to this magnetic circuit (OP circuit).  As the gear spins, the air gap reluctance \$R_g\$ changes, which changes the magnetic field for the coil, which induces a voltage in the pickup coil.  

Basic Faraday's Law.  No spin, no change.  Rotation produces a square wave proportional to tooth speed.  

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Air is a poor conductor of flux. So when tooth is close to sensor, the magnetic circuit is strengthened. When the air gap is large, the field weakens. Either way the flux experienced by the coil changes, Faraday's Law.

Take your initial circuit.  It is a magnetic circuit.  The magnet creates the magnetic flux \$\phi\$.  This flux is applied to coil with an iron core (flux conductor), the air gap and the toothed gear (flux conductor).

The tooth is iron (ferromagnetic).  Iron conducts flux about 5000 times easier than air.  So when the air gap is small the net flux becomes stronger.  The reluctance due to the air gap decreases.

Tooth rotates.  Air gap increases.  Air gap reluctance \$R_g\$ increases.  The net flux decreases.

Faraday's law says that whenever the flux linked with or associated with a circuit changes, a voltage is induced in the circuit.  

The coil is a circuit (connected up to a sensor).  Flux changes and a voltage is induced in the coil.  Repeat.  Square voltage waveform will be produced in the coil sensor as coil rotates.

Stop rotation.  No change in flux.  No induced voltage.