"There are people who have too much or too little of what makes us one or the other sex. " I think you are confusing transgender and intersex. Basically intersex means you do not fit the exact definition of XX female/XY male, e.g. a female XXY (Klinefelter). Transgender on the other hand , is properly defined by the GP, and people in such case really have gender disphoria, and is a category in the DMS V.
The issue is certainly confused by the fact that in some case, male were operated at birth and had gender reassignment, are sometimes called colloquially transgender, when properly it would be intersex, as they are missing (often) the psychological gender disphoria. One of the most well known example was a kid which had his penis removed at birth (botched circumcision) was tried to be raised as a female, but never felt like it and finally suicided at about 40 (John/Joan case - David Reimer).