There is no authoritative answer to this. The origin of the rule that an should come before an initial ‘h’ lies in the tie between English and French after 1066. In French initial ‘h’ is never aspirated (pronounced). So the French word hôtel is pronounced ‘ôtel’. The English upper classes followed the French pronunciation, and so would say ‘an otel’ with the ‘h’ silent. This habit persisted till well after WWII. But since the early sixties it has been dying out, so that today it would seem old fashioned and a bit posh. Much the same might apply to humiliation. But ‘a hot day’ has always been ‘a hot day’ with a haitch and a house has always been a house. These words are descended from anglo-saxon, not French.
So your instincts are right. But the use of ‘an’ before a no longer silent is moribund and should be allowed to pass quietly: a hhhumiliating defeat for tradition.