The record to which you link is unlikely to be a contemporary record of Nelson's birth. Without seeing the original documentation, we can't tell how faithful the transcribed data is.
Either way, birth records (from pre-1900 at least, in my experience) don't always contain full data on the documented individuals.
Without being able to view original documentation (basically primary authoritative records like birth-, marriage-, death-related records), we really can't tell one way or the other if Nelson Gray and Horatio Nelson Gray represent the same person.