Which AC bridge circuits are oscillators?
Please note that there is no "WIEN bridge". In 1891 Max WIEN (not WEIN) has invented a complex and frequency-dependent voltage divider with a bandpass characteristic R1, C1, R2, C2). This WIEN block was extended by ROBINSON using a fixed resistive voltage divider (R3, R4) to form a bridge (WIEN-ROBINSON bridge). 

At a certain frequency the bridge is tuned and the differential voltage (error voltage) between both of the bridge legs approaches zero. This is the case for R1=R2, C1=C2, R3=2R4). To form an oscillator we need an operational amplifier with very large gain to amplify this tiny differential voltage (for feedback purposes, resistive divider connected to the inv. opamp terminal).

However, in most cases we allocate both voltage divider resistors (R3, R4) to the opamp. As a result, we have an opamp with negative resistive feedback which is identical to a fixed gain amplifier. Thus, we have a circuit consisting of a WIEN circuit which is connected to a fixed-gain amplifier (gain of 3 for R3=2R4)  - and the amplifier output is fed back again to the WIEN block. This forms the well-known WIEN-osillator.

However, it should be mentioned that - by exchanging suitable elements of the tuned bridge - it is possible to derive three additional bridge oscillator configurations from this classical WIEN topology.