its pretty simple:-
z=2 after the comma suggests that the z coordinate of the point on the line is always =2 irrespective of the x and y coordinates!!
Lets simply things...:)
we take x-2/3=y+1/-2=z-2/0 (as your teacher told u ) =λ(say)
thus x=3λ+2,y=-2λ-1 , z=2 which does not hamper the given data!
but if we take z-2/1 then though we get the same x and y coordinates in terms of λ, z becomes=λ+2 that is z is not =2 which contradicts your given data
thus it will be z-2/o :)