The legend that Disney made reference doesn't have a title; however, it is about the goddess Hi`iaka (younger and most loved of Pele's sisters) and the young chief Lohiau from Kaua`i. And your suspicions are right, they did severely tame down the story to make it suitable for children. There are many versions of the legend...I'll share with you the version I was taught:
During a long slumber, Pele left her body and traveled to Kaua`i where she met and fell in love with Lohiau. Before she left Kaua`i to return to her body on Hawai`i, she told Lohiau to await her messenger. Pele asked her sisters to fetch Lohiau for her...none of Pele's other sisters would go on the journey, so Hi`iaka volunteered to fetch Lohiau.
Hi`iaka's best friend Hopoe lived in a sacred grove of ohi`a trees in Puna. Knowing Pele's fiery temper and jealousy of Hopoe, Hi`iaka knew that Hopoe would be at risk while she was away. Therefore, Hi`iaka made Pele agree that she would not harm Hopoe or the grove while on the journey. Afraid that Hi`iaka may fall for Lohiau, Pele demanded that Hi`iaka would not sleep with Lohiau and agreed that she would not she would not harm Hopoe or the sacred grove for no more than forty days.
Impatient for Hi`iaka to commence the journey, Pele forced Hi`iaka on her way without food and water. Hi`iaka only had time to grab a magic pa`u skirt that had the power of lightening. Hi`iaka climbed down the crater, chanting a song of forgiving love. Hearing this chant, Pele relented a little...giving Hi`iaka a portion of her divine powers as well as Pa`uopalae (who cared for the ferns around Kilauea crater and a woman of supernatural power) to be her traveling companion.
Hi`iaka traveled the long journey from Hawai`i to Kaua`i and encountered many obstacles along the way. Upon arriving at Kaua`i, she found that Lohiau had died from his deep longing for Pele. Hi`iaka revived him and they made the journey back to Hawai`i During the journey back, Hi`iaka and Lohiau found themselves falling for one another. However, Hi`iaka suppressed her feelings due to her promise to her sister.
Due to the obstacles that Hi`iaka encountered on her journey, the trip lasted much longer than the forty days. Convinced that Hi`iaka had slept with Lohiau, she sent a lava flow into Puna, which destroyed the grove and transformed Hopoe into a block of lava rock.
Hi`iaka, who had faithfully completed her task, found that Pele had destroyed Hopoe and the grove. In rebellion, Hi`iaka acted on her urges and embraced Lohiau, declaring that they were now husband and wife. Enraged, Pele summoned up an eruption, which did not affect Hi`iaka (who now had the powers of a goddess) but killed Lohiau.
Hi`iaka searched for and found Lohiau's spirit, freed his body from the crust of lava, healed Lohiau's body and returned his spirit to his body. They returned to Kaua`i, where they lived as chief and chiefess until Lohiau's death. After his death, Hi`iaka returned to her place among Pele's family.
If you ever want to read up on this legend some good sources are:
Hawaiian Legends of Volcanoes by W.D. Westervelt (full text is available for free at http://www.sacred-texts.com/pac/hlov/)
and
Pele and Hi`iaka by Nathaniel Emerson (http://www.amazon.com/gp/product/0976450801/ref=sr_11_1/103-7456948-5791861?ie=UTF8)