Actually in the example you cited, both are unambiguous in Standard English. When there is an ambiguity, pronouns follow the order of introduction in the previous sentence, if it makes sense to do so.
John and Bill met at the airport. He invited him to his home.
Is a slightly confusing way of saying that John invited Bill to his home (and never the reverse). It is valid (but confusing) because "John" is the first noun in the first sentence, and this matches gender and location to the "he" in the second sentence. Note that when the order is violated by making one of the pronouns unexpected and explicit, the sentence becomes invalid:
John and Bill met at the airport. He invited Bill to his home
John and Bill met at the airport. He invited John to his home.
In the case where you wanted to say that Bill invited John to his home, you would need to disambiguate:
John and Bill met at the airport. Bill invited John to his home.
(Note that the "his" in the second sentence again refers to Bill).
There is no such confusion when the pronouns are different:
Bill met Mary at the airport. He invited her back to his house
(Mary was invited to Bill's house)
Bill met Mary at the airport. She invited him back to her house.
(Bill was invited to Mary's house).
Confusions arise when nouns are introduced not all in the same sentence.
Bill travelled to New York on Tuesday. David travelled there on Wednesday. On Thursday, he invited him for drinks.
Is another example of an ambiguous (and hence wrong) sentence.
My advice, however, is that when the sentence becomes complicated it's always better to err on the side of caution, and to be explicit avoiding pronouns.