If your meter measured the (average) value of the Current and multiplied it by the (average) value of the Volts, you would get an answer which corresponds to the so-called VA. This is useful to know but doesn't tell you the Power you are consuming.
Multiplying the instantaneous values of V and I and then averaging, will tell you the Energy consumed. Electronic meters can do either, quite easily, using the values of actual I (from a current transformer) and V .
I think it's an historical thing. The rotating disc meter actually does this instantaneous multiplication sum which produces the Energy consumed. The regulations and the billing has been done based on this information, which sounds like just what you want. But it ignores the problems (and costs) associated with poor Power Factors. There are requirements for large equipment users to keep their Power Factor reasonable in order to limit the extra losses which are associated with V and I being out of phase. I believe you have to pay on the basis of Max VA, in some circumstances, because this has implications on the spec of the supply equipment.