  (Abridged) We report the detection of X-ray emission from the host cluster of
the radio-quiet quasar H1821+643 with the ROSAT HRI, and the non-detection of
the host cluster of the radio-loud quasar 3C206 using the EINSTEIN HRI.
CL1821+643 has a rest-frame 0.1-2.4 keV luminosity of 3.74$\pm$0.57
h$_{50}^{-2}$ 10^45 ergs/sec, 38% from a barely resolved cooling flow
component, which places it among the most X-ray luminous clusters known. The
cluster emission complicates interpretation of previous X-ray spectra of this
field; in particular, the observed FeK$\alpha$ emission can probably be
attributed entirely to the cluster, and either the quasar is relatively X-ray
quiet for its optical luminosity or the cluster has a relatively low
temperature for its luminosity. We combine these data with the recent detection
of X-ray emission from the host cluster of the `buried' radio-quiet quasar IRAS
09104+4109, our previous upper limits for the host clusters of two z$\sim$0.7
RLQs, and literature data on FR II radio galaxies and compare to the
predictions of three models for the presence and evolution of powerful AGN in
clusters: the low-velocity-dispersion model, the low-ICM-density model, and the
cooling flow model. Neither of the latter two models can explain all the
observations. We suggest that strong interactions with gas-containing galaxies
may be the only mechanism needed to explain the presence and evolution of
powerful AGN in clusters, consistent with the far-IR and optical properties of
the host galaxies studied here. However, the cooling flow model cannot be ruled
out for at least some objects, and it is likely that both processes are at
work. Each scenario makes predictions for future X-ray and optical observations
which can test their relative importance.
