I'm going back to your original post, because I think I can help relate the kinematics to the force in a little different way that might work for you. The tension in the spring is always going to be T = kΔL, where ΔL is the change in length, relative to the original unextended length. If ΔL is negative, then the spring is in compression, and the tension is negative.
Let xL represent the x coordinate of the left end of the spring at time t, and let xR represent the x coordinate of the right end of the spring at time t. Let the original unextended length of the spring be L0. So we are going to allow the possibility that both ends of the spring can move, and not just one end. So the left end can move by moving your left hand, and the right end can move by moving your right hand. With this description, the change in length of the spring ΔL relative to its original unextended length is given by:
$$ΔL=x_R-x_L-L_0$$
It doesn't matter which hand is moving and in which direction, the only thing that matters is the distance between your left hand and your right hand, xR-xL. The tension that each hand feels is then given by:
$$T=k(x_R-x_L-L_0)$$
If the term in parenthesis is negative, then the spring is in compression.
Hope this helps.
Chet