Why The three classic feedback oscillator topologies are Colpitts, Hartley and Pierce. A Colpitts takes its feedback from a capacitive voltage divider across the tank, a Hartley takes it from a tapped inductor, and a Pierce is essentially a Colpitts that uses a quartz crystal as the resonant element, which is why it dominates crystal oscillator design. All three meet the Barkhausen requirement: loop gain of at least 1 with 360 degrees (in phase) feedback at the desired frequency.
Watch out The other choices sprinkle in invented names like Taft, Fenner and Beane; negative feedback is also a giveaway, since oscillators require positive (regenerative) feedback, not negative.
CHP: Colpitts = Capacitors, Hartley = Henrys (coil tap), Pierce = Piezoelectric crystal. Any other surname is fake.