the difference, at least in this case, is the fact that "cal" could refer to a horse in general and you don't have to say anything more about it. on the other hand, "calul" asks for details about it so we would know about what exact horse you are talking. ex: eu stau pe calul care mănâncă = i stay on the horse who is eating. or eu stau pe calul meu= i stay on my horse. there are some situations where you could keep your sentences "unfinished" without problems. ex. îmi plac merele.= i like apples.
I think the use of "on" instead of "onto" in this context is pretty common, and search numbers in Google and Ngram would suggest even more so:
"It was Good Friday, 120 years ago. A man limped out the back door of Ford's Theater, jumped on his horse, and sped down darkened Baptist Alley"
New York Times