"This holiday is designated as 'Washington's Birthday.' Though other institutions such as state and local governments and private businesses may use other names, it is Federal policy to always refer to holidays by the names designated in the law."
Washington's actual birthday was February 22, and that's when the holiday was originally celebrated. However, in 1968, Congress passed the Uniform Monday Holiday Act , which designated the third Monday in February as Washington's Birthday, beginning in 1971.
The Presidents Day moniker evolved later, at least partly to honor Abraham Lincoln, whose birthday was February 12, but some hoped it could be a day to honor all presidents.
Of course, states, local governments and businesses are not bound to honor the Uniform Monday Holiday Act, so they can call Monday whatever holiday they like or not have one at all.