Or just buy a hose end water meter for $20So if you can be assured that they actually deliver what they say they deliver (5000 gals per truck, or whatever), then you can use the meter trick.
Right before you take over the filling, find your house's water meter and take a pic of the reading. Start your fill hose(s). When the water level gets halfway up the skimmer opening, stop the fill and go out and read the meter again. The difference between the two numbers plus the trucked-in amount will be your water volume. If they're close, and you're close, you should get a pretty decent number, probably way better than the PB's estimate.
You can get a bit closer if you minimize water use in the house while the hose is on: turn off landscape irrigation, minimize showers and flushes, postpone laundry, etc.
If they want some water in the pool before they dump the truck's water, or if you use the hose between trucks, however, then you just keep track of the meter readings and hose use. You're not supposed to let the water level sit in one spot for any length of time during the fill. It can create a permanent ring-stain at the point you paused the fill. So you might want to have the hose running between trucks. However it happens, just track the meter while the fill hose is running and then add it all together when the pool finishes filling.
This might be the only opportunity to do this, because you may never have the pool completely empty again.
Yeah, your house meter can do the job, but for $20 you can see the amount of water your hose is putting out at a glance and not have to worry about irrigation systems to toilets or anything else.