Actually, Volvie, the current walls of the old city are not the original Tanach-era walls. The current walls were mostly built by Suleiman the Magnificent (1500s). The location of the current walls do not match those of the original walls. Ir David, for example, where Dovid Hamelech lived, is outside the current walls.
More importantly, I don’t know if there is any relevence whether one is within the walls or not. For example, all of Yerushalayim reads the megilla on Shushan Purim. (According to some, the halachic definition may not exactly match the municipal boundaries, but that has nothing to do with east/west.)
Finally, most of “East Jerusalem” is also outside the walls. As everyone knows, the distinction between West Jerusalem and East Jerusalem is only political, in that East Jerusalem is the part of the city that was occupied by Jordan from 1948 until 1967.