What are our earliest museums? Well, the Handbook of Texas begins its article by Ron Tyler on museums ....
"MUSEUMS. Museums are a relatively recent development in Texas. Perhaps the first museum was nothing more than the exhibition in a local hotel room of an itinerant limner's portrait samples, such as Thomas Jefferson Wrightqv presented in Houston in 1837. But in 1879 Sam Houston Normal Institute established a museum to preserve materials relating to Sam Houston,qv hero of the Texas Revolution,qv president of the Republic of Texasqv and senator and governor of the state, and in 1881 Albert Friedrich began the collection of horns that became the Buckhorn Hall of Horns in San Antonio (see BUCKHORN SALOON). By the turn of the century the Museum of Human Anatomy at the Medical Branch of the University of Texas (1890), the Strecker Museumqv at Baylor University in Waco (1893), and the teaching museum at Our Lady of the Lake College in San Antonio (1896) were in operation, and art associations throughout the state had given birth to fledgling art galleries in Houston, Dallas, San Antonio, Fort Worth, and several other cities."
Read more about it http://www.tshaonline.org/handbook/online/articles/MM/lbm3.html
So even before 1900 we had history, natural history, science, and art the four basic categories? The topic could be interesting scratch on. The Online Handbook cites "museum" being used at least 500 times. (As an aside, before TSHA relied upon Google to search, more than 500 hits could be retrieved and were presentable in more than little 10 citation page units. Oh, the good old days.)