I think they are the same person (wherever he was born) ... not perfect English but you know what I mean.
These are the facts as I see them-
There is a
submitted entry on the IGI that Herbert was born at Highland Castle, Carlisle. This entry has no date, no baptism. Furthermore, there is no such place. The parents are named as John HILLS and Anna REYNOLDS ... if these
were correct, they may have married at Westminster in 1834 (transcribed entry, not submitted). This same John HILLS is alleged to have died 10 Dec 1848 (submitted entry).
Herbert A HILLS seems to have attended Eton followed by Oxford University before marrying Anna GROVE in 1863 (in London). Interestingly (perhaps), Anna had sisters born in Italy and France (1851 census), so her family got around a bit too.
As I said, there was Edmund born (possibly) in High Head Castle, Cumb in 1864. This place was about 8 miles south of Carlisle ... I doubt it would be described as "Carlisle" in any
official record. It is close enough, I'll hope you will agree, to be the Highland Castle mentioned speculatively in 1837?
In passing, I noticed that there was a John HILLS bap in Paris in 1828 (son of John and Mary Ann) ... could he have been a half brother of Herb's? If so, it seems that John HILLS (senior) may also have done a bit of globetrotting (diplomatic service?).
Perhaps Herb was also involved in such colonial matters. I see he was missing in 1881 and you say he wasn't around in 1901. I suppose one must wonder if he was still actually married to Anna by 1881 in view of the apparent lack of young children.
One further point - the page about High Head Castle - that I linked to - said it was built mid 18th century and the family was squabbling for the next 100 years over it. Unless it can be shown that John WILLS was linked to the BROUGHAM family, it seems unlikely that he was living there in 1837.