I think one of the big problems is that like many other medical terms Intersex is a very broad term which covers a vast variety of different conditions all of which can have subtly different outcomes and effects on a persons development of self identity.
And indeed the line between transgender and intersex is not as well defined as some in both communities would like to believe. I myself existed for many years precisely on that borderline with, to some extent, a foot in both camps. It was only when I had to have some complex medical investigations at the age of 47 (for a problem the details of which I won't go into) that the evidence finally emerged which settled the issue and proved that I had in fact been intersex all along. Up until then, although I had had some suspicions I would have said I was Transsexual.
By the time I had the evidence it was academic anyway as I had already gone through the full transition and surgical proceedures nearly 30 years earlier!
But being intersex involves so many different conditions. Just look it up in a medical directory. So it isn't surprising that some of us (like me) feel that we are one gender or another, and others of us don't. Neither is wrong. Likewise some of us elect for surgery, some us don't, again neither is wrong.
The only things that are wrong are, when society tries to force or bully someone into compliance with a set of rules for which they are not well suited, or when some doctor has the arrogance to presume to mutilate someone before they are old enough to make their own internal identity choice clear.
I would always have chosen female, but others make a different choice or indeed do not acknowlege that they should have to make a choice at all.
So the really important thing is that society allows EVERYONE whether, intersex, trans or just boring old vanilla, the freedom to be whatever they feel comfortable being.
I suppose what I am saying is that as humans we are often far to keen on labels and categories and rules. But the fact is nature is infinitely more variable in it's expression than our narrow definitions ever allow. So rather than trying to label and understand a better way is to just accept and embrace one another.