I think you're thinking about it too much. Programmers have been saying "of type integer" or "of type boolean" since I learned to program decades ago. It didn't evolve from the use of italics or anything like that. They didn't have fancy formatting in program comments.Basically, the class and the type used in that sentence means the same but it isthe Horse class comparing to of type Halter where the definite article is missing. Why?
My guess would be that authors have often used a technique where a word is printed in italic type dropping the definite article before the word. After times this evolved to the when authors at times didn't even care about making a word italic but kept dropping the article. Could you please correct me if I am wrong.
I would imagine that since "Horse class" is the subject of the sentence, it gets an article, where when it is used as an object, it doesn't. I bet you can find somewhere a sentence saying that something something is of class Horse.
Student or Learner