Definition of the offset voltage for a single-ended op-amp
If I rigorously apply the definition of the offset voltage, I should find that voltage which, applied between the two inputs, sets the voltage of the output node to zero.


For a lot of cheap op-amps, the offset voltage and the voltage amplification ratio are both high enough that the range of differential voltages that keep the output in the linear region is dominated by the offset voltage.

For any decent datasheet* the test conditions will be detailed; i.e., they will tell you what the output voltage was, and possibly the input common mode voltage (see, e.g., TI's LM324 and TI's LF353).

* In the "only true Scottsman" sense.