Sorry, this isn't clear to me. Why "du" when we are talking about one horse? Shouldn't it be "le"?
So can you somehow see difference in this phrase if it is about specific horse or random, uncountable one if it is fixed?
You're right, but du is a contraction of de + le. 'Pres de' is a single phrase, and since it's followed by 'le cheval', it becomes 'du'.
Interestingly I saw "pres du" in another sentence and typed "next to" and it told me I was wrong and it needed to "near the." This time I put "near to" which may be somewhat incorrect but it said that the correct answer was "next to." Can "pres du" mean "next to" or not?