You are making good progress. The problem with using RMS values is that unless the load is purely resistive, you must know the power factor for the load you are driving. This is usually unavailable or so inaccurate as to be useless.
As you can see from the current waveforms for the non-resistive or non-linear loads you are driving, they are quite different from sinusoidal and if you will display the voltage waveform at the same time you will see that the phase relationship is no longer constant either.
True power is the (integral over time T of i(t) * v(t) *dt) / T. Where i(t) and v(t) are the instantaneous current and voltage. You can approximate this integral pretty accurately by sampling both the instantaneous voltage and current simultaneously at a high rate by latching the values with separate sample and hold circuits and doing an a/d conversion.
The meters the power companies use don't measure power correctly either which is why if the know you have loads that are very reactive, they will apply a "fudge factor" to the rate to compensate for the error.
That's probably more than you really wanted to know about it but if you are interested in knowing how accurate ac power measurements are made in the lab there is a good article that doesn't assume you have a math degree that explains the process pretty well.