Confusion on State Space Equation of Buck Converter
The two state variables \$x_1\$ (the inductor current) and \$x_2\$ (the capacitor voltage) do not change directions between the on- and off-states. You can see that during the off-state, as correctly drawn in your lecture notes, the inductor current still splits between \$C\$ and \$R\$.

If I may further comment, I would recommend to use the PWM switch model rather than the state-space averaging (SSA) technique for analyzing switching converters. See here for more details: slides 46/47 for the buck converter SSA derivation and the rest of the pages for the PWM switch model analysis.