Bench mill the head moves up and down on the column, and often the head may also tilt side to side to work at an angle. The term bed mill may also be used this style of mill, usually those on the larger end, I guess because they are to big to fit on a bench.
A knee mill has the table move up and down instead of the head. The head can usually tilt side to side, and many can swing (pivot on the column) and extend allowing it to reach more of the table, or even off the table. Some also can nod the head tilting in or out. The Bridgeport mill is the classic example of a full featured knee mill.
Mill drill is kind of a vague term. Often applied to round column mills, but also used for many square column bench mills. Round vs square refers to the column the head attaches to. Round allows the head to swing, but generally has less positive control of the head position so requires the user to adjust more frequently. Square uses dovetails to control the movement of the head up and down, so has more positive control, but trades some flexibility in movement and costs more.
Square column is generally preferred, but Round has some benefit.