For BIBO stability, the poles cannot lie on imaginary axis?
Yes it is true and there is no contradiction between these two definitions of stability.

The s-plane is a representation of:
$$s=\sigma + j\cdot\omega$$

The left half of the s-plane mean $$\sigma&lt;0$$

The imaginary axis is given by $$\sigma = 0$$ because in that case all you have is 

$$s= j\cdot\omega$$