No, again, intent is whether the action was deliberate or not. IOW:
Intent: Did she intend to shoot the person? (this can be self-defense, not self-defense, warning shots, etc, motive does not matter one bit to answer this question). As spurraider said, it doesn't even need to involve shooting, just intention to do bodily harm is enough.
Motive: What was the rationale for the intent?
Premeditation: Was motive pre-planned?
But, then, again, all 3 go into charging and setting up a case. In home invasion, self-defense cases, intent could be relatively irrelevant, for example.
The main reason to establish intent is to determine if the death was voluntary or involuntary, and this doesn't always necessarily matches the accused's claim. They could claim they fired warning shots, and forensics might come back with irrefutable evidence that's not the case. So at trial, the prosecution will prove intent based on that evidence, etc. That then backs up the charges they filed.