Correct way to measure power consumption in AC
The three types of power relate as follows:

Real power - power that does work.
Reactive power - power that does no work, but nevertheless circulates back and forth to the source
Apparent power - vector sum of reactive and real power

Your instantaneous current measurement will be giving you the sum of reactive and real (resistive) current. Multiply this by the voltage at the same instant and you get power at that instant.
The trick is, reactive loads have a +/- power polarity, while resistive loads are always positive or zero. When we sum these, whether we are computing averages or tallying energy, the reactive part cancels out, leaving real power as the result.
For example, we compute the average:

avg. real power = [V0I0 + V1I1... + VnIn] / n

Over the sample set, on average all the reactive power will be equal to zero.
Here's a concise list of formulas for all these:

From here: https://blog.teledynelecroy.com/2018/03/power-calculations-for-distorted.html
And a more scholarly treatment of the topic: http://engr.case.edu/merat_francis/eit/PE%20Review%20Chapter3.pdf