Why there is many configurations of Colpitts oscillator?
If the active device is a transistor (BJT or FET) there are, in general, three different amplifier configuratios possible: Common emitter, common collector, common base.

The first two references show a common base Colpitt oscillator (base on signal ground).
The last ref. contains a Colpitt oscillator in common emitter configuration.

Feedback path:

(a) Common base: The transistor output is connected to an LC resonant block which is high-resistive for the desired frequency. The feedback signal is sampled between both capacitors (cap. divider) and is in phase with the output. This is necessary for positive feedback because in common-base configuration there is no phase shift between emitter and collector. The ratio between both capacitors determines the amount of feedback.

(b) Common emitter: The working principle is somewhat different. The collector is connected to a grounded third order lowpass (r,out-C1 and L-C2). The output of this lowpass has a phase shift of -180deg at the desired frequency. This is necessary because there is a phase shift of another 180deg between base and (correction): collector (in common-emitter configuration). Hence, again we have positive feedback.
(Note that r,out is the transistor`s output resistance at the collector node).