We use $^2$H NMR to study the rotational motion of supercooled water in silica pores of various diameters, specifically, in the MCM-41 materials C10, C12, and C14. Combination of spin-lattice relaxation, line-shape, and stimulated-echo analyses allows us to determine correlation times in very broad time and temperature ranges. For the studied pore diameters, 2.1-2.9 nm, we find two crossovers in the temperature-dependent correlation times of liquid water upon cooling. At 220-230 K, a first kink in the temperature dependence is accompanied by a solidification of a fraction of the confined water, implying that the observed crossover is due to a change from bulk-like to interface-dominated water dynamics, rather than to a liquid-liquid phase transition. Moreover, the results provide evidence that $\alpha$ process-like dynamics is probed above the crossover temperature, whereas $\beta$ process-like dynamics is observed below. At 180-190 K, we find a second change of the temperature dependence, which resembles that reported for the $\beta$ process of supercooled liquids during the glass transition, suggesting a value of $T_g\!\approx\!185$ K for interface-affected liquid water. In the high-temperature range, $T\!>\!225$ K, the temperature dependence of water reorientation is weaker in the smaller C10 pores than in the larger C12 and C14 pores, where it is more bulk-like, indicating a significant effect of the silica confinement on the $\alpha$ process of water in the former 2.1 nm confinement. By contrast, the temperature dependence of water reorientation is largely independent of the confinement size and described by an Arrhenius law with an activation energy of $E_a\!\approx\!0.5\ $eV in the low-temperature range, $T\!<\!180 $K, revealing that the confinement size plays a minor role for the $\beta$ process of water.