Laplace tranfsorm of capacitor functions and initial conditions
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When solving circuits using Laplace transform, one method commonly taught is to replace a capacitor with an initial voltage with a capacitor with zero initial voltage and a special voltage in series with it. Slide 25 and 26 give an example. The equivalent circuit in \$s\$ domain has a capacitor \$C\$ with impedance \$1/(sC)\$ and a voltage source \$v(0)/s\$ in series. This equivalent circuit has the same characteristics as the original capacitor.

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For AC circuits, the AC impedance is given as \$dv/di\$ (ratio of change in voltage to change in current). A similar argument is used to arrive at the formula \$1/sC\$ in the Wikipedia article for impedance
As such, I do not see any problem in expression you have derived; It's just that the initial condition is handled separately and impedance of capacitor is taken as \$1/sC\$. This turns out to be easier to work with in practice.