As I understand it the "open" in the BOFRA title dates from the days when there were both professional and amateur codes of fell running. All runners were able to take part in BOFRA races whereas only amateurs were allowed to enter FRA registered races. Any "amateur" running in a BOFRA race risked being banned from FRA races.
It is still the case that anyone can enter a BOFRA race but only BOFRA members qualify for the championship. This is similar to the FRA championship rules where an unattached (i.e. not a member of an athletic club affiliated to UKA) runner must be a member of the FRA to count.