There are two commonly "blamed" parts which are said to cause a "proud cut" horse. The term "proud cut" really is a behavioral term meaning stallion-like, but there is a physical part to it too. A TRUE "proud cut" colt retains TESTICULAR tissue; the vet/castrator botched the opperation so badly as to leave a chunk of testicle behind.
The two often-blamed parts are the epididymis and the vas deferens. The vas deferens is a tube running from the testicles to the end of the penis. It connects the organs together and facilitates transport of both semen and urine out of the body. It is JUST a tube, nothing more. It in no way produces or causes any behavior.
The other part is the epididymis. This is an organ which is located right next to the testis. It is ALMOST impossible to remove the testis without removing the epididymis but it is possible. However, this organ is a holding and maturing area for sperm. It does not produce any hormones and can not in any way influence behavior. If a vet was to disect off the organ and leave it behind on purpose, all it would do is give the colt a few more days of ejaculating sperm, even though they will be dead.
In summery, a proud cut horse is a horse which shows stallion like behavior, 99% of the time it has no physical basis regarding a botched castration unless it was botched BADLY. Don't go blaming other organs, and if you must, blame the adrenal gland. Its the one producing the hormones.