So if the load is inductive and the waveform on it is similar to a sine wave, this calculator will not be able to take measurements correctly?
Yes. switching loss is a function of how Ids and Vds overlap during switching. These shapes will depend on the impedance of the circuit being switched (i.e. resistive, inductive, etc). The formula in your calculator is valid for pure resistive loads (where Vds and Ids change simultaneously). For inductive loads, Vds and Ids overlap more,
see figures 2 and 4 in this document. This causes switching losses to be higher than with a resistive load.
In theory it is possible to measure the rise/fall time of Vds by observing the gate waveforms, but only if the miller plateau is clearly visible (refer to those same figures in the above document). But in your case the miller plateau cannot be seen at all. So instead you should actually measure Vds directly. As for Ids, it's very difficult to measure directly (especially at high switching frequencies). Usually the rise/fall times of Ids are calculated from device models or simulation.
You mention the waveform is a "sine wave," is this sort of resonant circuit? In that case neither the restive nor inductive load equations are valid for you.