With her new memoir My Name Is Barbra, Barbra Streisand says she hopes readers will finally “know the truth” about her.

“One of the reasons I wrote the book is to talk about the myths about me,” Streisand tells CBS News’ Gayle King in an interview that aired on CBS Sunday Morning.

Among reflections on her career as a singer, actress and director, Streisand writes about her “past loves and regrets,” as King teased.

But if it was up to Streisand, married to James Brolin for 25 years, there would have been less about her past relationships — including with Don Johnson, Ryan O’Neal, Andre Agassi and former Canadian prime minister Pierre Trudeau — in the book.

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“Listen, I didn’t want to write about any of them,” Streisand says. “[But] my editor said, ‘You have to leave some blood on the page!'”

After King complimented the Way We Were star on her “very nice dating roster,” she asks Streisand if she had “a good time.”

“With the men in my life, yes,” she replies.

Elsewhere in her sit-down with King, Streisand, who is joined by Brolin at one point and recalls the apparent insult that sparked their courtship, reflects on her childhood, including being “angry” about the death of her father when she was just a year old and her mother’s lack of affection.

Despite her iconic career, Streisand also insists, “I don’t think of myself as a famous person, I just don’t. I’m the same Barbra Joan Streisand as I was in high school.”

The multi-hyphenate also tells King about the work she finds most fulfilling: “I love directing, because it’s a complete vision … It calls on every aspect of yourself: decorating, visualizing, helping actors achieve a performance that I have in my mind for them.”

Watch the full interview, in which King also tours Streisand’s private mall in the basement of her Malibu house, below.