The information on the tire sidewall is a statement of the maximum load capacity and the cold inflation required to provide that load capacity.
Different tires have the word "max" placed in different parts of the phrase which leads to the confusion.
Here is a post on passenger tires.
Here is a post on when the Minimum inflation is the Maximum.
Note: A a statement of Max Load "At" stated inflation such as "Max load 3,000 lbs at 80 psi" is leaving off the implied "minimum" before the 80 psi.
Also introducing the word "cold" seems to confuse some.
Maybe this will help:
Tire inflation should only be set when the tire is at ambient temperature and has not been warmed by being in sunlight or driven on for the prior two hours.
A tire may be loaded to the number of pounds stated on the sidewall, only when the tire has been inflated, when cold, to the stated pressure on the tire sidewall.
The problem is that these phrases don't fit in the same place as current "Max Load 3,000 lbs at Cold inflation 80 psi"