Is the Wien Bridge oscillator essentially a resonant filter?
Before Johnson noise ever kicks a Wien bridge into action, the sheer monstrous transient of applying power  does the job.


  Some schematics show a Wien bridge oscillator using a tungsten lamp as
  a source of resistor noise.


It may look like that but almost certainly (as certain as I can be without the schematic you saw) the lamp is there to provide non linear gain so that the peak to peak amplitude doesn’t crash into the power rails.


  This seems quite different from other oscillators which use the rising
  voltage on a capacitor connected to the DC power supply to build up
  oscillations.


You seem to be describing a Schmitt trigger single RC oscillator. It doesn’t use phase shift at all. It uses thresholds and the charging time of the capacitor to dictate oscillation frequency. Go look up relaxation oscillators.


  Part of this question is based on the fact that neither the Wien
  bridge oscillator nor the phase shift oscillator have inductors.


These type of oscillators rely on the changing phase shift of a network of resistors and capacitors to produce a phase shift of 180 degrees at one particular frequency. At different frequencies, the phase shift is not 180 degrees and hence, when wrapped around an inverting op-amp, won’t oscillate.

In fact oscillators like the Colpitts (that use an inductor) hardly ever oscillate at resonance because the phase change at perfect amplitude resonance just isn’t right for sustained stable oscillation. Taking this further, hardly any (if any at all) oscillators use perfect amplitude resonance. Try googling Barkhausen stability criterion for oscillators.

Whether the oscillator is Wien bridge or phase shift or, "so-called" resonant oscillators like the Colpitts, Hartley or Pierce, the guiding principal is that of providing the right phase shift to produce a stable oscillation.