I think Jack's lovely, though I think Henry Edward scans slightly better than Jack Edward. And yes, Harry has traditionally been a nn for Henry, it doesn't seem at all odd to me that you'd have one as the formal name and one as the nn (though ditto for John/Jack...).
Sorry you're getting so much flak for the popularity thing; I had the same on a girls' name thread and chickened out of putting any of our boys' names up for discussion as I didn't want to risk being talked out of them!
The thing is, there are so many more names in circulation than there used to be that even the top name (Oliver, in the most recent stats), was only given to 8,427 babies in 2010. Out of 797,000 babies born that year. (Though yes, I realise half of those were girls! In other words roughly 10% of boys born that year are Olivers). All I'm saying is that compared to the days when fully 50% of babies were called either John or Mary, choosing even a very popular name doesn't condemn your child to a lifetime of anonymity.
And I disagree that a name is just a utilitarian mode of identification - that implies that every parent should invent an entirely new name, or spell an old one in a made-up way. It can also be about linking your child to its family or heritage/honouring someone who means something to you etc etc, and having a familiar name roots you to a particular culture, which I don't see as a remotely negative thing.
Sorry, I'll stop banging on! Anyway, I think your names are lovely, and you can't go wrong with either of them.