Which cultural museum in Oklahoma is dedicated to preserving and interpreting Native American history?

The National Cowboy and Western Heritage Museum primarily focuses on celebrating the history of the American West, cowboy culture, and western art, rather than specifically on Native American history.

In contrast, the Cherokee Heritage Center is dedicated to preserving and interpreting Native American culture, specifically that of the Cherokee Nation. It features exhibits that educate visitors about the traditions, history, and contributions of the Cherokee people, making it a vital institution for understanding Native American history in Oklahoma.

The Oklahoma History Center, while it includes a range of Oklahoma's history—including Native American history—serves a wider focus on the state's overall heritage, integrating various aspects of cultural narratives.

The Philbrook Museum of Art, located in Tulsa, is primarily an art museum that showcases a diverse array of visual art rather than focusing specifically on Native American history.

The Cherokee Heritage Center stands out as the museum specifically aimed at preserving and interpreting Native American history, particularly that of the Cherokee, making it the correct answer to the question.