A thirteen yard load is quite large. In fact if its really 13 yds its most likely overweight on most roads unless the dump truck had extra axles. If you had some kind of "box" that is 3'x3'x3' to dump into, that would be 1 cubic yard of dirt.
The easiest way is to deal with reputable suppliers.
The next solution is to observe the delivery and take a quick measurement of the dump box in feet. (LxWxD)/27 will give you the Loose volume in cubic yards.
A little less accurate is to measure the pile and do some math. The angled sides makes the math a little more complicated.
The issue you will have with trying to measure it with your bucket, is you will be compacting it somewhat when you fill your bucket. So the Loose yards delivered will compute to less volume in your bucket.
From a construction standpoint, soil is in one of three states:
Bank - as it sits in undisturbed ground.
Loose - as it was transported in the truck.
Compacted - after it is placed and compacted with equipment.
For typical soil, 1 bank yard = 1.25 loose yards = 0.9 compacted yards.
The reality is most soils are sold by weight, not by volume. It is the only way they can accurately buy and sell the product. The issue is most people can calculate how much they need by volume and are not aware how to convert that volume to the volume in the state they need it and then convert that to tons.
The worst I have seen was getting compost. We pulled in and paid for the level volume of the dump box. We pull up to where the loader operator was working. This place is huge, probably 20 acres piled about 16 ft high. The guy in the loader drives into the pile, scoops up a bucket, backs up and shakes it out of the bucket to fluff it, then scoops the fluffed compost and dumps it into the truck. Only loads the truck about 2/3 full and parks the loader. I walk over and tell him what we paid for. He tells me his bucket holds x amount and that he loaded what I paid for. I counter with the volume of the box and that not only is he short, the fluffing routine shorted it even more. I go back to the counter and complain and they won't do anything about it. I should have just demanded my money and dumped it back in the pile. Needless to say, I never bought anything from that supplier again.
Found another suppler and I always got more product than I paid for.