We just got our 2 cats vaccinated today and under the short hair female they have "DMH" and with the long haired male they have "DSH". What do these to terms mean?

DSH - domestic short hair
DMH - domestic medium hair
DLH - domestic long hair

HOWEVER, the medium hair is technically a long hair cat even if its not as long as a Persian or Maine Coon. In cat shows you only have long or short hair classes, no medium hair - those cats are considered to be long hair cats.

Cats of no particular breed / unknown ancestry are classified as "domestic" and further broken down by coat length. Domestic Shorthair and Domestic Longhair are the most common terms used. but some people make a 3rd category for Domestic Medium Hair. Genetically they are actually longhairs, but the longhair gene has variable expression.

DSH = Domestic Short Hair pets question and answers,www.5d2d.com
vs.
DMH = Domestic Medium Hair

The length of the hair is the difference.

My Cats told me.

Domestic Short Hair vs Domestic Medium Hair DLH would be Domestic Long Hair.
And the tags must have been switched