This convolution problem involves the convolution two signals of finite duration. You would expect that the duration (extent) of the resulting convolution be the sum of the durations of the two signals. Also, you would normally expect that the convolution result would have three non-zero intervals of interest:
In the lowpass RC circuit, the initial condition is the initial voltage across the capacitor at t = 0-. The initial condition is zero. The system passes the all-zero input test. And the system is LTI.
When a rectangular pulse is the input signal x(t), the convolution can be computed over three intervals of interest: no overlap, partial overlap and complete overlap.
When viewing the circuit behavior, the capacitor charges when the input pulse is "on" and discharges once the input pulse has turned "off". The system response to a rectangular pulse input is non-zero from t = 0 to t = oo.