The official name of the federal holiday remains "Washington's Birthday," though of course to complicate things further, some states do have an official Presidents' Day holiday -- or some variation of that, see below -- which happens to be the same day as the federal holiday, the third Monday of February.
The holiday is "President's Day" in Alaska, California, Colorado, Hawaii, Idaho, Maryland, Nebraska, New Hampshire, Ohio, Oregon, Tennessee, Washington, West Virginia and Wyoming. It's "Lincoln/Washington/Presidents' Day" in Arizona; "George Washington's Birthday and Daisy Gatson Bates Day" (who?) in Arkansas; and "Presidents' Day" in Hawaii, Massachusetts, New Mexico, Oklahoma, South Dakota, Texas, and Vermont. Maine calls it "Washington's Birthday/President's Day," while it's "Presidents Day" in Michigan, Minnesota, Nevada and New Jersey. The holiday is "Lincoln's and Washington's Birthday" in Montana; "Recognition of the birthday of George Washington" in North Dakota; "Washington and Lincoln Day" in Utah, and "George Washington Day" in Virginia.
It should also be noted that the third Monday in February can never be February 22, the New Style date on which the adult George Washington celebrated his birthday. The third Monday can only be from February 15th to the 21st, which explains the anomaly in some years of having Washington's birthday celebrated on a Monday a week ahead of his birthday (the 15th) rather than a week later on his actual birthday.
One more thing: George Washington, son of Augustine and Mary Washington, was actually born on February 11, 1731, Old Style. After the adoption of the Gregorian calendar by the British Empire in 1752, Washington changed the date to be mathematically correct, and the year to conform to the new custom of starting the year on January 1.
Also born in February: Ronald Reagan (6th) and William Henry Harrison (9th), and of course Abraham Lincoln (12th), who is honored on his birthday with a holiday in a few states. Woodrow Wilson died on February 3 and John Quincy Adams died on the 23rd.
Among Vice Presidents of the United States, the dead Aaron Burr and Henry Wilson were born in February (the 6th and 16th, respectively). The living Dan Quayle celebrates his birthday on February 4. Vice President Charles Curtis is the only veep to die in February, on the 8th.