There are several types of proofs. If you want a general idea, look for them in the Scott U.S. Specialized catalog.
The first distinction is between "die" and "plate" proofs. A die proof - of any size - is made from a single die, usually to test the die or to offer a variety of colors from which the issued stamp will be printed in. These latter are often called trial color proofs.
The large die you are showing in the link is a die proof. This is a single print. If you look at the card it was printed on, you can see the squarish outline of the rest of the die as it was impressed into the card. Often, these are impressed into a sheet of india paper (very thin, very absorbent and great for die impressions) which has been laid atop the card. The pressure from the printing impresses the design into the india paper and presses the india paper into the card.
Plate proofs are made from a plate of images, the same as a sheet of stamps is. The distinction here is that these impressions are most often done on india paper or thin card. There are multiple images printed at the same time.
To call the item shown a large die plate proof is an error by the seller. If it is a single image proof, it is a die proof. If there are multiple images, it is a plate proof.
Here are some examples to give you an idea:
This first one is a large die proof, reduced to the size shown. Note that this item still shows the entire area which was impressed by the printing into the card. This is Scott 66P1. While it may look similar to Scott 65, the color is not the same.

Next is a plate proof on card. This would have been cut from a sheet of the plate proofs to create the individual proof. This is Scott 65P4.

A very nice collection of plate proofs can be obtained still and makes a great alternative to a collection of mint 19th century items. The BEP sold these in five different printings in the 1890's. The entirety of the issues created up to that time was offered at face value (these were without varieties such as grilled issues or color variants).
There were some other printings later, but that might complicate things for now.
Let me know if you have any questions.