It is a matter of conventions.

I defined the initial tension as positive, thus when the rod is pushing the mass upward and the sliding block downward. Since at t₁ the rod is pulling the mass and the sliding block, thus the tension is negative. Similarly, at t₂ the net force is such that the rod is pulling the mass. So the tension is negative too.

I should have noted that somewhere in my reasoning. Thanks for pointing it out.