Electromagnetic wave propagation in time-varying media offers unique possibilities for controlling wave dynamics beyond spatially structured systems. While both nonideal step transitions and the suppression of reflection in the adiabatic limit are well-established phenomena in the WKB framework, these analyses have predominantly relied on the approximation method and assumption of electric displacement ($\mathbf{D}$) continuity. In this work, we provide a comprehensive comparative study of wave scattering by establishing exact analytical solutions for a linear temporal transition under both the standard $\mathbf{D}$-field continuity and the electric field ($\mathbf{E}$) continuity boundary conditions. The latter is explicitly motivated by realistic experimental scenarios such as capacitor switching in transmission lines. By utilizing Bessel functions to construct closed-form solutions, we reveal that while the scattering coefficients in both scenarios exhibit similar trends of damped oscillations as the transition duration increases, their specific amplitudes and momentum-exchange behaviors differ significantly because of the distinct boundary mechanisms. Furthermore, building upon the established WKB framework for soft temporal switching, we derive analytical approximations for both boundary conditions. By benchmarking these results against our exact Bessel-function solutions, we provide a precise quantitative definition of the adiabatic regime through a normalized switching threshold ($\tau /T_0$), thereby elucidating how specific boundary physics dictates the validity range of adiabatic approximations. All analytical derivations are validated against full-wave numerical simulations, demonstrating excellent agreement. This rigorous analysis not only serves as a precise mathematical benchmark for defining the validity range of adiabatic approximations but also elucidates how physical boundary conditions fundamentally dictate the electromagnetic response in time-varying systems.

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