In the Bible itself, it says straightforwardly that the extra heh that was added to
Abram's name came from the word "hamon", meaning many, referring to his fathering many nations. That's also what it says on this site's entry. I'm pretty sure the whole "heh from God's name" thing came up much later as part of a folk etymology in some of the interpretations.
Anyway, the added heh only makes sense when added to the M that already ended his name, because it gives the name a whole new meaning. This means it is definitely not the same element as in
Aharon (which is probably not even Hebrew in origin).
Jaron is from
Yaron - totally separate name. Baby name books are not always correct. Yuds do not randomly disappear from Hebrew spellings of a name, especially when they are part of the element yah.