Many horsess have white hair/fur, straight pure white, but as mentioned, they weren't all born this way, other than horses of colour. The reason a "white" horse is referred to as grey I believe is to distinguish between the albinos and "white" horses (many if not most are born darker, also mentioned above). I could be wrong here but I think albinism was not ever common but was more just a bit more frequently occuring in history back.
Also, the difference in terms could relate to the skin colours of each horse. The "grey" having black skin and an "unofficial" white being pink. Or not... : )