The magnetometer in your phone is a MEMS sensor which measures mechanical deflection of a current-carrying element. The deflection is caused by the Lorentz Force, i.e. force induced by an electron current flow in a magnetic field (in this case, the earth's magnetic field).[1] The magnetometer in the linked article senses (EDIT: corrected, hopefully) oscillation in the magnetic field of protons, a result of Larmor Precession[2]. Remarkably, the oscillation frequency is proportional to the ambient magnetic field strength, and the frequency is in the audible range. The circuit works by rotating protons in the fluid so that their magnetic axis align, this results in a synchronised bulk magnetic field oscillation that is large enough to be sensed by a simple tuned amplifier circuit.[3]

Further, the magnetometer in your phone is a 3-axis device that measures the orientation of the magnetic field, whereas the magnetometer in the linked article detects only the strength of the magnetic field (in fact, is tuned to detect only a single strength/precession frequency).

[1] https://en.wikipedia.org/wiki/MEMS_magnetic_field_sensor

[2] https://en.wikipedia.org/wiki/Larmor_precession

[3] https://en.wikipedia.org/wiki/Proton_magnetometer