I was told, that axanthics lack both in most snakes, but anerys only lack red.

This is basically correct. Axanthism is defined as the lacking of yellow pigmentation but it also can include red pigmentation as well. Anerythrism is basically a subset term of Axanthism and is defined as only lacking red but can and will exhibit yellow pigmentation if it is present.

Anery is lacking red. Axanthic is lacking yellow. With hogs its appears axanthics lack both red and yellow, so they are both.

Your statement here is actually contradicted by the above statement you made in the same post. Hognose cannot be both Axanthic and Anerythristic because if it was Anerythristic, then it would have to be exhibiting yellow pigmentation which it is not. Being that Axanthism already is defined as lacking yellow and sometimes red pigmentations, it is the only term that is relevant to hognose.

In other species, there are anery's(lacking red) that have yellow, cornsnakes for instance.

Correct, and again, if hognose were in fact Anerythristic, they would exhibit yellow pigmentation which they do not.

To put it simply, all Anerythristics are actually Axanthics, but not all Axanthics are Anerythristics.

I will refer to the definition stated by Dr. H. Bernard Bechtel who was a Dermatologist. In Dr. Bernard Bechtel's book Reptile and Amphibian Variants he states quite clearly (page 65):
"Axanthism is a hereditary defect of xanthophore pigmentat metabolism, resulting in absence or decreased amounts of red, yellow, and intermediate pteridine pigments.".

He even went so far to call Anerythristic corn snakes Axanthics because Axanthism helps define the specific subset term Anerythrism and eliminates confusion caused by both terms (page 66):
"Because of the predominance of red pigmentation in their pattern, corn snakes are also called red rat snakes. Their red pigment is synthesized by xanthophores, the same pigment cells that synthesize yellow. When xanthophores fail to function, neither red nor yellow are synthesized, but since the wild-type corn snakes are so overwhelmingly red, the absence of red in axanthic specimens is their most conspicuous feature. For this reason, they are called anerythristic (without red). While anerythrism is perfectly correct, I prefer to call all snakes with this mutation axanthic. This describes the mutation, and avoids stumbling over what to call tricolored kingsnakes with this mutation, since they clearly are both "without red" and "without yellow".".
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Troy Rexroth
Rextiles

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