One way to get a feel for it is to look at Tesla's web site for their charging calculator, which is here:
Home Charging Calculator
In essence the website says that:
A Model S with a 48 amp charger can charge at 29 miles/hour if the source supplies 40 amps, and at 52 miles/hour if the car has the 72 amp charger and the source is a 90-amp circuit. There are also values for other charger configurations and (lower) values for Model X. Model 3 might be different.
Those values of 8 kw (not kwh) and 16 kw for the destination chargers are a bit confusing, I think. But I agree with you that that is how they are listed. (I have also seen 13 kw for a few places.) I think the values might depend somewhat on when the chargers were installed as well as how large a circuit they are on.
My own Tesla Wall Connector is on a 50-amp circuit so it can charge at a maximum of 40 amps. If the voltage is nominally 240 V, that would be 9.6 kw, and the single charger on my car is rated for that amount. I think that is probably the lowest rating Tesla would have allowed for a destination charger. So I suspect the 8 kw figure is a conservative version of the 9.6 kw rating, to reflect possible lower supply voltage. For my Model S, the 40 amps from my home WC adds roughly 30 miles of range for each hour of charging.
Tesla's web site says that the Wall Connector can provide up to 72 amps for cars equipped with the appropriate charger, so at 240 volts that would be 17 kw. But older cars that were equipped with the second 40-amp charger could expect to charge at 80 amps or 19.2 kw.
Guessing here but I imagine what Tesla has done is to standardize on the 8 kw and 16 kw figures as representative of more realistic voltages and that they represent 40-amp or 80-amp configurations. Or possibly 40-amp and 72 amp. Although I have charged at Tesla destination chargers a couple of times, I can only charge at 40 amps so I may not have hit the local limit.
Hope I have not confused you more than I helped!