Robotic Manipulator 3-RRP v2

I recently asked a question about a 3-RRP manipulator but the only answer didn't work for my robot.

I would describe the manipulator as an arm that rotates around the shoulder over the x-y axis, the elbow that rotates around the x-y axis also and the forearm - hand that moves FORWARD and BACKWARDS. (check attached image for a more described explanation).

The solution I calculated cannot be correct as the "hand" moves up and down and not forward and backward.

I tried setting other values to the DH-parameters but I still think this is not the correct answer.

My professor said that we are not allowed to use a structural parameter to rotates the axes and that the use of 3 rows on the DH-parameters is the correct solution.

I've been trying to get this for almost 1 week now and seems impossible to me.

On the image you can see some of my attempts to solve this problem.

A detailed answer with the reason for the values of the DH-parameters would be awesome.

In your new attempt, the transformation of axes from origin-1 (i.e. $$x_1, y_1, z_1$$) to origin-2 ($$x_2, y_2, z_2$$) is incorrect as $$x_2 \not\perp z_1$$ and $$x_2 \not\cap z_1$$, violating DH convention.