With a moon in Libra, we would expect Harry to be kind, considerate, and of passive temperament in his early years. This aspect also suggests steadfast and helpful companions--for example, the faithful Hagrid, Ron, and Hermione. As a name, Hermione is another word for the god Hermes or Mercury, and Hermione does function like Harry's sun conjunction to Mercury--knowing all the incantations and rituals required for the successful execution of magic. (In Book 3, "The Prisoner of Azkaban," we find that Hermione is almost certainly a Virgo.)
These three planets, however, are in a square to Uranus in Scorpio. (A square
is a right angle relationship of tension and stress.) This correlates with the series of magical contests where Harry is pitted against powerful and unpredictable forces of the Dark (that is, Scorpio), represented by Malfoy, Crabbe, Goyle, the Slytherins--and "You Know Who."
Harry's life completely turns around on his 11th birthday as he finally finds out that he was born of wizard parents, that his mother and father were killed by a powerful sorcerer on the Dark Side, and that he is to begin his training at Hogwarts School of Witchcraft and Wizardry.
For the 1979 natal chart to apply, there would have to be important astrological transits occurring in 1990. (Transits are particular relationships between the positions of planets at different times.) Such is the case. Near Harry's 11th birthday, in the spring and early summer of 1990, transiting Saturn was square his moon, suggesting the despair he faced at the repression of the Dursleys. But, in addition, transiting Neptune trines Harry's Saturn throughout the last half of his 11th year, indicating the opening of his true destiny. This long-term aspect was exact at the time of Harry's birthday, a prime symbol for the beginnings of occult and spiritual studies. Given these considerations, I would conclude that July 31, 1979, gives the best chart for Harry Potter.
An alternative (the next July 31st on a Tuesday, in 1984) should be given some consideration, however. Interestingly, this chart works pretty well, too. Here the sun in Leo is in trine to Uranus and square to Scorpio, suggesting the same themes of Light versus Dark, access to magic and higher mind, and a destiny to lead. And, remarkably, in the spring before this 11th birthday (1993), transiting Saturn is opposite Harry's moon, a very similar aspect to the one that occurred for the 1979/90 chart! In passing, I'll note that many of the gods and heroes of antiquity (for example, Venus) are "twice-born," and thus have two birth charts.
So, do we have Harry Potter's true horoscope? And is Harry Potter real? That's for you to decide.