According to my study, "good at something" means "can do something well"
and "good at something" means "can use something well".
So, "She is good at English words" might mean she know a lot of English words
and "She is good with English words" might mean she is good at manipulating or using English words
Is that explanation acceptable?
We use "good at" with a skill, that is something you can do, and do skillfully:
She is good at programming. She is good at tennis. She is good at English.
But it wouldn't make sense to say "She is good at cats" (for example) because you can't "do cats".
So "She is good at English words" is rather odd. It doesn't mean anything. It doesn't mean that she knows lots of English words.
On the other hand you can say "good with" a tool or something that you can use, control, or guide skillfully
She is good with a pen. She is good with children. She is good with cats.
So it would be correct to say "She is good with English words". It means that she is skillful at using English words.
