Did all Popes use the inverted cross? The above image shows Pope John Paul II sitting on a chair with an inverted cross on it. According to tradition, the inverted cross symbolizes Saint Peter.

Was Pope John Paul II the only pope who used the inverted cross publicly?
I don't know more context for the picture, but the title "Pope visits holy site of Sermon on the Mount" indicates, that the chair is not a part of Pope's usual attributes, but that it's part of the chapel on Mount of Beatitudes.

This inverted or Saint Peter's cross is not very common in catholic context, but very few Catholics would feel bad if they were on the same picture with such a cross. I have seen the same design as is in the picture in few other places, not connected with any pope. It is used mostly of aesthethic reasons, the artist usually don't care whether the cross is Greek or Peter's, but they like it to be connected with the edge of the item it's engraved to, and cross of St. Peter is ideal for this.

By the way, more than with Peter's cross, pope is associated with Papal Cross.