I worked at a stud farm, not the national stud, but at a regional stud, in the construction of new stables, and i was given a good look around behind the scenes while working there..
Your correct in saying that they use a teaser (of a mare in heat), to arouse the stallion, but once the stallion is aroused, he is led to a fake mares rear end... complete with a cushioned vagina (i kid you not!!!) and is encouraged to mount, at the other "end" the stud vet collects the semen from the said stallion..which is then artificially inserted into the mare...all this should i add, is supervised by the mares owners, after that monies are paid on conception and birth of foal.
PS..Seem to me the national stud "educational tour" is sexed up a little for the tourists!!!