I Young's Modulus

If you look at the equation for stress: F/A, a thicker rod would have more stress for the same force. Thus it would bend less without breaking.

This is just using the equation, I don't know why.

 

For the same bending angle, the stress is less in the thinner bar.
The breaking stress is a material characteristic so it is the same for two bars made from the same material.

 

Extensions and compressions are also less in a thin rod. They are proportional to the distance from the neutral axis.
But how can be "wrong" to discuss it in terms of stress?

 

You haven't really explained why the stress is lower in the thinner rod with just that answer. You should really mention that the stress is lower for a given bending angle because the strain is lower, and this is because of the distance from the neutral axis.

 

Because the bending stress depends on the distance between it's neutral axis (which is in the center for common shapes in the case of pure bending). So a thicker rod will experience larger maximum bending stress than a slim one when subject to forces of the same magnitude.

 

same reason why a arc of small radius subtends a larger angle at center than a arc of same length and larger radius