Step 1: Start from the meaning of the word cyclic: the coordinate itself is absent from $L$, even though its velocity $\dot{q}_i$ may appear. Absence from $L$ is expressed mathematically as the partial derivative with respect to $q_i$ vanishing.
Step 2: Write down what each option means physically. Option (B) $\partial L/\partial \dot{q}_i = 0$ would kill the conjugate momentum, which is not the definition. The two $\dfrac{d}{dt}(\cdots)$ options are consequences or parts of the equation of motion, not the defining property.
Step 3: Only $\partial L/\partial q_i = 0$ states that $q_i$ does not enter $L$ explicitly, which is exactly the definition of an ignorable coordinate.
Step 4: As a check, this immediately gives a conservation law $p_i = \partial L/\partial\dot{q}_i = \text{const}$, a hallmark of cyclic coordinates such as an angle for a central force.
\[\boxed{\dfrac{\partial L}{\partial q_i} = 0}\]