(Hopefully) quick grammar question: I am having difficulty figuring out when I am supposed to use なくて vs. when I am supposed to use ないで. I can try to remember by grammar point, for example if you are requesting someone not do something, you use ないで ください and if you want to say it’s ok to not do something, then you use なくてもいい, but it’s challenging to remember every instance. Is there any general rule, or do I have to buckle down and remember every individual case? Is there a difference if it comes after an adjective, noun, or verb?
For linking sentences together, なくて is used when you’re just tying two clauses together so that it’s not two sentences, and this can have the typical nuances of て to merge clauses.
Using ないで to connect clauses means “do [following action] without doing [preceding action].”
So, 走らなくて入った would mean, “(Subject) didn’t run and then entered.”
走らないで入ったwould mean, “(Subject) entered without running.” In other words, it describes the way they entered the room.
ずに is more formal/literary than ないで. Other than that, they have the same meaning. It’s not impossible to use ずに in spoken language though.
ず with no に is typically equivalent to なくて, just more formal/literary again. But it can’t do things that have て nuance implications, like implying causal relationships.
And it’s also possible for ず to be like ないで, but I don’t recall more of the details off the top of my head.
As a follow up question, if you aren’t talking about an action but rather a state of being, which do you use? For example, this is not an apple, it is an orange instead: これはりんごじゃなくて、みかんです。I think this matches with your explanation because you’re just linking two clauses together, but am I correct in thinking that, or does the implication that you’re correcting something false change what you say? If not, could you also link adjectives the same way, for instance, これは 白くなくて、黒い。?