Companies renting dump trucks typically rate the dump truck by the
neat (struck)volume as described below:
If the dump truck bed is 8'6" x 16'-0" with 36" sideboards, the truck
would be be called (nominally) a 16 cubic yard truck.  [(8.5 x 17 x 3)
/ 27)] = 16 cy.

However, it is recognized that a loaded truck does not hold the same
yardage as its nominal designation.

What is needed is a set of industry recognized published data that
defines the load factor typically used for estimating purposes for a
dump truck hauling soil (sand/silt/clay mix) if the hauling payment is
to be "by the load".

You mean, normally, or actually, the load is not exactly equal to the
nominal volume of the truck?
For sand, silt, and/or sand/silt/clay mix materials, that is
understandable or obvious. If the truck were exactly full of those
fine aggregates, then some of the material will spill over the top of
the walls of the truck's bed during travel time due to bumpy roads,
sloping roads, sudden stops, wind. The canvass or whatever protection
placed on top of the material during travel time cannot prevent
spillage by bumps/slopes/stops.
Even againts the wind, the cover over the material cannot provide 100%
protection. (If you've been behind such a truck on the road, you know
what I mean.)
So, the trucks are purposely loaded a little bit less than their
actual/exact capacity by volume. The top of the material is a few
inches below the rims/tops of the bed walls.

Spillages of such material is dangerous for the other vehicles on the
road. The truck driver will get ticket from the highway patrols.

For larger aggregates, those that don't easily slide over each other
at bumps/slopes/wind, the load can even be more than the exact volume
of the truck.

If you rent the trucks "by the load" and you are not comfortable on
the less-than-the nominal volume delivered, then you adjust your
calculation. Say, a lload is 97% of the nominal. (You can compute
that. Determine the average depth of the top of the delivered material
from the tops of the walls. Divide that by the actual depth of the
walls from the bed, convert that to percentage, then subtract that
fromm 100%. If the cross section of the truck's bin is not
rectangular, then use some more Math to calculate the actual volume
deivered, compare that to the nominal to get the percentage
delivered.)

I see all this trouble is related to the volume delivered/hauled. You
are particular with the volume delivered, aren't you?

Depending on the volume you want delivered, What if you rent the
truck/trucks by the day, or per day, per 8-hr-a-day day?
If it is on a regular basis, what if you buy your own truck and hire
an operator for the job?

Am I assuming too much?

Firstly a nominal yardage is offered as a aproximation, the volume of
a truck bed not being a critical mesurement indeed if piled high it
could deliver a higher volume. I would have thought the weight the
truck could carry would be the first concideration and then depending
on the cargo the bed size would be the next concideration, posably why
it is quoted as a nominal size, as it is not the limiting factor. The
industry standard would in the case be the cubic yard, if you require
a 3" coverage over a 28 square yard area you would order and estimate
in cubic yards, it matters as much that the load may be 5% under or
over as it would that a random sample of ten spots all mesured exactly
3". When making a mix the industry standard is a shovel !. I think the
answer is that the industry standard is a mesurement that over meany
years has been found it fit the product. To apply mathamatics to a
load of sand could be seen as overcomplicating an industry where the
value is not in the materials.

Whats wrong with the industry standard is that its based on volume
rather than mass. It would be much fairer to pay by the ton rather
than the yard.
Haulage of construction materials or demolition/debris is generally
constrained by weight. Therefore if you are doing demolition on T&M it
would be much better if trucks were weighed leaving the job. Obviously
a different rate per ton would apply for contaminated clay rather than
re-cyclable concrete.
The density of most building materials is well known.
E.G. Steel 490Lb/ft3; Concrete 149Lb/ft3; Clay 120Lb/ft3; Sand
115Lb/ft3; Water 62Lb/ft3
Your 16CY truck if you could theoretically filll it to the brim with
non spillable "material X" will probably only be allowed carry an 18
ton payload anyway. The density of "material X" would be
18x2000/16x27= 83.3 Lb/ft3.(Somewhere between water and sand).
18 tons of 4" thick concrete floor slab on the other hand would occupy
732 SF  241.6ft3 or 9CY.
If is much easier to calculate the number of loads on a concrete
demolition job or a soil offhaul job by weight and divide by the
allowable tonnage per truck to calculate the number of loads. The
weight doesn't change-the volume can.
In general,for most cases the weight is the limiting factor rather than the volume.
This argument admittedly does not cater for loads like trees/brush,
and rough demolition.