From that website, look near the bottom and they only list one cedar; Spanish Cedar. Click on that and it takes you to woodfinder.com. On that site, look down the list and find Aromatic Red Cedar. When you click on that, it states, "DESCRIPTION: Very fine textured and moderately hard for a softwood species."
While Paxton Lumber "names" it as a hardwood, they do not "mean" that it is a hardwood -- only that the wood is relatively hard. They also list multiple pines as hardwoods. If you look at Balsa and Basswood, for example, they are classified as hardwoods, but are two of the softest woods known to man.
There are many other hardwoods that are softer than the hardest softwoods.
Here's a website that gives a nice, detailed explanation on the differences between the two, and what the criteria is for each:
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