"Miriam," "Maria," and "Mary" are three forms of the same name. "Miriam" is closer to the original Hebrew. "Maria" is the Latinate form. "Mary" is the Anglicized version of "Maria."
In Christianity, St. Mary is the mother of Christ, and revered by all Christians, but particularly Roman Catholics. "Mary" conveys a sense of identification with Christianity. In the United States, there are no rules, and people give their children any names they like, often without reference to cultural or ethnic identity, and, to quote the words of a 1906 song, "Mary is a grand old name." Nevertheless, a Jewish woman would be more likely to be named "Miriam."
"Mary" used to be an extraordinarily popular name for women. When I was growing up in the 1950s and 1960s, and for many decades before, it was the most popular woman's name in the United States. Rather surprisingly to people of my age, it began to decline and now is no longer even in the top 100. In fact, the form "Maria" is now more popular than "Mary."
Apparently the meaning of "Miriam" is unknown, but in any case it doesn't matter much. In English-speaking culture, we don't pay a lot of attention to the meanings of names. People may look them up when they are choosing a baby's name. But they are barely present in our thinking. When I introduce myself as "Daniel," nobody thinks "God is my judge." They are more likely to wonder if I was named after Daniel Boone or Danny Thomas or Danny Kaye.