As FEA Way mentioned, the couple of keywords *DAMAGE INITIATION and *DAMAGE EVOLUTION are used to define progressive damage. Subsequently, it is adapted to ductile materials. In your first example, I would still refer to the first drop as the onset of damage. However, given that you have a fragile type of failure, I would question the validity of the "progressive damage" framework.
In your second example, the orange arrow corresponds more of less to the elastic limit in my opinion. I'd take the green one as the onset of damage. Be aware though, that you have to use the plastic strain, not the total strain, when using the keyword *DAMAGE INITIATION with the ductile or Johnson-Cook criterion.