Hermaphrodite is generally considered a curse, please do not use it. It is a stigmatic word hermaphrodite mean that the individual has male and female genitalia and both work and reproduce
which as far as we know it is impossible in humans!
What did usually happens is someone to have ambiguous genitalia a person with a vagina that has Clitomegaly (a super clit developed that resembles the penis, both have the same embryonic origin), a cavity between the scrotal sac resembling a vagina, or another type of genitalia to stay in the middle ground between the penis or the vagina
The intersex goes beyond that, it can be a chromosomal variety as XXY among others, in gonados as presence, in the same individual, testicular tissue and ovarian tissue in any combination: + ovary testis (laterally); ovary or testis ovotestis + + ovotestis (unilaterally); bilateral ovotestis (bilaterally)
Ovotestis: gonads with testicular both aspects as ovulatory
Basically any physical characteristic or genetics that can not be framed in "male or female"
Intersex people may or may not have gender identity not binary, This same rule applies to people who present no intersex, intersexuality For apply to Morphological and Chromosome features
Morphological: external appearance of matter