Ghoul mostly refers to members of I. vulgaris, especially domesticated sub-species, called I. vulgaris vecare that are the majority of the ghouls working in the ministry.
I. vulgaris vecare has four main breeds: Fire, water, air and earth.
Multi ghouls usually are used as an umbrella term for ghouls without clear classification, but is mostly a descriptive than a valid scientific term. It can refer to wild members of genus Infernide, most often I. vulgaris or even feral I. vulgaris vecare that are too mixed to have one clear dominant element.
Wild members of the genus lack the clear deviate of the base elements.
Quintessence ghouls however are I. via lactea vocare and lack as diverse breeds as I. vulgaris vecare. Mostly we have quint, nether, aether and void breeds. But they mostly have an diverse aesthetic and magical differences. It's much rarer to summon a purebred one.
All members of genus Infernide can (usually) interbreed, with their offsprings having ability to further reproduce. However many ghouls have clear preference towards their species, sometimes even breed due to some species/breeds being not the most compatible for natural breeding.