To begin I will start with what is known that is 100% true and scientifically accurate and correct.
There are many different phenotypes of dwarfism, which means there are many different types of dwarfs that have different problems and physical abnormalities.
That being said, there are obviously many different mutations in different genes causing these different dwarf types.
With that you need to consider which type(s) is/are the most prevalent and each sample of each type being identical in its abnormalities to be consistent in it phenotype.
OK, now the most prevalent type I have seen and documented is a type that is very similar to cattle dwarfism, which shows symptoms and phenotypes consistent with a human type of dwarfism called achondroplasia. This type of dwarfism in humans by chance (or maybe not) is also the most prevalent type of dwarfism in humans as well.
Now in order to be an achondroplasic dwarf in humans you must have one parent that is an achondroplasic dwarf or be an achondroplasic person born by two normal parents from a spontaneous mutation in a bone growth gene, I will spare you the specifics, but this type of dwarfism is called a dominant genotype, which means that if you have even only one copy of the gene you show the disease, so a human with Dd heterozygous genotype is a dwarf, dd is a normal person, and DD is a homozygous genotype dwarf, which is worse than Dd.
Now for miniature horses it is different, a dwarf of achondroplasic characteristics, is born to two parents that look like normal horses. This is what is called a recessive genotype, which means you can have the mutated gene and not show the disease. This means that a horse that looks normal can carry a mutated gene recessively and not show the disease characteristics, or phenotype. So, in order for a dwarf to be born, both horses, sire AND dam MUST carry the gene to produce a dwarf. THIS IS A FACT for a recessive disease gene to show its phenotype, both the sire and dam are carriers of the diseased gene, and BOTH passed their copy of that diseased gene to the foal that shows the phenotype of the disease. So it takes both the sire and dam to make a dwarf.