I would go with the intent being the deciding factor. I have OD'd in the past without intending to die, but just to get a break via being unconscious. I have also come close to dying via self harm, but it was never intended to kill me.
I classify attempts as something deliberate that I had hoped would kill me (premeditated or impulsive). I don't count the self-harm taken too far, because I was not wanting to die, just gain relief (and feed the addiction - more was needed to get the same high/relief from it).
If your cutting was done in hopes of killing yourself (even if the wounds would not have been sufficient to do it), I would classify that as an attempt, even if it was not planned. If you had not intended to die from it (Or even wished to die from it), but came close to dying, I would just put that in the category of a self-harm slip-up and not an actual attempt.
I'm glad you are working with a therapistnon everything. I too understand wanting to classify stuff. It makes some things easier to understand.