You may have been okay with using the names of actual races, although it could depend on how you portray the races. Changing the names of the horses was a good idea, I think. Overall, I think changing all the names will make things easier and simpler on you.
Although if you're going to upload the story in full on the internet then you might have trouble getting it published. Publishers generally require 'first publishing rights', and uploading stories onto the internet is considered publishing. One chapter will be okay though.