What I'm getting at is how do we know that there were not several types of horned god.

I'm pretty sure the main horned god of my part of Yorkshire had goaty horns, but I know of other types of horned images being found in the region - rams horns for example. I am of the opinion that these are two different god images for two different gods.

During the time of these early celtic gods, England and France were covered by great forests. The stag, with its great tree-like antlers, sheds them in the autumn and then regrows them in the spring, this would be seen as a renewal of life. It would have represented the spirit of the forest, its speed and agility respected by hunters. Fertility also, its great roar when fighting for females must have made it an awesome creature. Its depicted on the Gundestrup cauldron, in one plate with Cernunnos, and on another where a god grasps a stag in each hand. Even today, when you see them move through woods, the female deer appear and disappear like magic.
They also figure in irish mythology quite strongly, mostly through shapeshifting...