It is exactly like you wrote, I have written longer explanation so you will understand all the nuances.
ないで can be used only with verbs. なくて, on the other hand, can be used with nouns, adjectives and verbs.
But if なくて is used with a verb (not in fixed expression), just like you wrote it has to express reason or cause. (ないで doesn’t have this requirement, also when なくて is used with nouns and adjectives it simply means “not, and/ is not…, but”)
**In other words, if you simply want to say: “not and” for verbs your default choice is ないで and for adjectives and nouns なくて。If you want to state a reason/cause then you can use なくて. (though it is usually used in certain fixed contexts, more here, in other cases から and ので are better)
In addition to that: ないで and なくて have their own set expressions in which they sound natural:
ないで is the only one that is used with auxiliaries like ください、ほしい、おく。
Plesae, do not do that.
I do not want you to say that.
Because I have pasta, I didn’t buy it (in preparation for something).
なくて is preferred in:
Verbなくて+ すみません (though ないですみません is more or less acceptable)
なくてもいい - don’t have to (like above, ないでもいい is also more or less acceptable instead)
忙いそがしいなら、行いか なくてもいい です。
If you are busy, then you don’t have to go.
Or なくては(いけない/ならない) (cannot be rephrased to ないで)
You have to eat breakfast.
There is also another expression, ずに, which is formal version of ないで (and like ないで cannot be used with nouns and adjectives).