I hope I got your point and try to answer your question. Actually, if I understand you right, then what you call attack can also be seen as an adversary acting in a specific attack model.
Therefore I briefly review the model for digital signatures.
Goal of an adversary:
We start by discussing the goals of an adversary beginning with the strongest and ending up with the weakest attack goal.
Total break: The adversary is able to obtain the secret signing key. Thus, he is able to impersonate the signer by signing arbitrary messages in the name of the signer.
Selective forgery: The adversary is able to produce valid signatures for some selected messages or a particular class of messages.
- Existential forgery: The adversary is able to produce at least one signature for a message, which has not been signed yet (the adversary typical has no control over the choice of this message).
Power of an adversary:
After defining the goals, we will take a closer look at the adversary and define his ability or power. Thereby, we start with the weakest and end up with the strongest one.
Key-only attack: The adversary solely knows the public-key corresponding to the secret signing key of the signer.
Known-message attack: The adversary has additionally access to a list of message-signature pairs from the signer, whereas he has no influence on the choice of the messages.
- Chosen-message attack: The adversary has access to a list of message-signature pairs, whereas the messages were chosen by the adversary before attempting to break the signature scheme.
- Adaptively chosen-message attack: The adversary is able to adaptively choose the messages which are signed by the signer during the attack. Thus, he may choose messages depending on the public-key of the signer and also previous messages resp. signatures, which were obtained during the attack. In other words, the adversary can use the signer as a signing black-box (oracle) throughout the entire attack.
This is not complete, as there are other types as random message attacks (but thats not important here).
Security as a combination of goals and power:
Now, the combination of the goal and the power of an adversary gives the type of security of a specific scheme. In general, a digital signature scheme is said to be secure if the most powerful adversary cannot even achieve the weakest goal.
In the notation introduced above, a digital signature scheme is considered secure if it is existentially unforgeable under adaptively chosen-message attacks.
Coming back to your question:
Actually, what you describe as an attack, is a total break under a key-only attack. Or you may see it as an adversary mounting an attack (key-only) in this model (total break).
Hope this helps.