Hermaphrodite is a relative term...
No, an enlarged clitoris does not make you one.
Rather, its the position and function of the urethra... If she pees thru it, then it will generally be termed a vestigial penis.
If not, its usually termed be a hypertrophied clit.
But the real defining issue is whether the person actually has Indeterminate or both sets of primary sex organs...
True Hermaphrodites usually have malformed or immature testes up inside the body...even though they externally appear as a girl... ( prone to going cancerous) or vestigial and malformed ovaries... or sometimes indication of both.
Occasionally a fully male child will have a dysfunction of testosterone production, which can result in a person with immaturely formed male genitalia, who nevertheless develops in other physical ways as a girl.
These people, who have small male genitals and breasts, and do not develop facial hair ...are what is usually meant by the term hermaphrodite...
though genetically they usually show up as boys, with an X and a Y.
Some unusual sexual development is caused by having 3 sex chromosomes... two X's and a Y or two Y's and an X.
Oddly... the Clit is actually formed very much like a tiny penis.. and the scrotal sack is made of the same tissue as a woman's inner labia... a bifurcated scrotal sac, with a very small penis can be visually indistinguishable from a vagina on a newborn...
The medical term is ambiguous genitalia... and it really can be a challenge to accurately assess the sex of some children born this way.
In the past 40 years... the medical community tended to perfomr an "assignment" of sexuality, based in part on how the doctors best thought to proceed... with the confident assumption that the proper hormones would "define" the person as a woman or a man.
However... long term studies have shown this idea to be deeply flawed...
Gender identity seems an elusive and multifaceted thing and the current advised course it to wait and see the gender the child tends to chose as they grow.
These kinds of cases are very strong evidence that primary sexuality is something determined at a very early age by factors unrelated to environment, upbringing and, even , hormonal influence.