Question

A repeated back-and-forth or upand-down motion

Answer 1

That depends on if the movement is rapid or not. Regardless of speed,  this is an oscillation, but if it is very quick then it is often called a vibration.

OR

This could be simple harmonic motion too. (See attachment for example).

Any of these is correct, but bear in mind that simple harmonic motion only occurs in specific conditions (i.e. the force on the object is proportional to the distance from where it starts moving).

Hope this helps!

Answer 2

a repeated back and  forth