It sounds like you are on the right track. You had a faster response from Lucy the second time around and it will only get better. Now that she sees that you will stop her from aggressing on the cats she understands that you have an opinion about her behavior and she will start to think about her manners before acting out.
The snap or your fingers should be sufficient from now on - it is a startle that catches her thought before it becomes an action and she reminded that she needs to rethink her attitude to the cats and the food
You are taking charge and teaching her what is permitted and what is not. The sooner you set boundaries and rules the faster she will settle in and be clear on what is expected of her.
Imagine moving into a new family as a child, would you rather the parents left you alone to figure things out or would you feel better if they took the time teach you the rules of the house and how to get along.
I don't care who eats first just so long as they eat with good manners.
Love Them & Lead Them,
~Elizabeth & Doug
Dog Training the Way Nature Intended