I think, the common classification is as follows:
1) Oscillators based on a BANDPASS as frequency-selective feedback path (zero phase at f=fo) require a positive amplifier in the loop (Example: WIEN oscillator). That means: Both parts (amplifier and feedback path) must not exhibit any phase shift at f=fo. Therefore they do NOT belong to the class of phaseshift-oscillators.
2) There are other oscillator types based on an inverting amplifier (180 deg phase shift). In this case, the frequency-selective feedback path must also produce 180 deg at f=fo in order to produce a loop phase of 0 deg at f=fo (Barkhausen condition). For this purpose, the feedback path is realized as a third-order lowpass or 3rd-order highpass circuit (Hartley, Colpitt, Pears, Clapp).
3) As a special case within the class of phase-shift oscillators, there are some oscillator types which produce the required 180 deg phase shift using two integrating stages (at least two additional opamps). In this case, the circuit produces at two different integrator ouput nodes two different signals, which are off in phase by 90 deg (sin and cos). Because these two signals are "in quadrature" these integrator-based oscillators are called "quadrature oscillator".
4.) A special case within this class of "quadrature oscillators" is the BUBBA-oscillator. Here, instead of two integrating stages (each with 90 deg phase shift) four passive RC-blocks are used (decoupled with a buffer) - each producing 45 deg. of phase shift at the desired frequency fo.
However, again it is possible to use two opamp outputs for providing at the same time two signals "in quadrature".