Something I just realised.
In English, you can say I have no dog or I don't have a dog, but in German, only the first is usual: Ich habe keinen Hund. That is, you walk about what you have (no dog) rather than what you don't have (a dog).
You could say "Ich habe nicht einen Hund", but that's marked, not neutral: I'd expect one of "Ich habe nicht einen Hund, sondern eine Katze" (I have not a dog, but rather a cat); "Ich habe nicht einen Hund, sondern zwei" (I have not one dog, but two); or "Ich habe nicht einen Hund" (I have not a single dog) -- each sentence with some manner of emphasis.
Similarly with, say, "I haven't got any money", which in German would typically be "Ich habe kein Geld" (I have no money); "Ich habe nicht Geld" parses for me not as "I don't have money" ("Ich (habe nicht) Geld") but as "I have not money [, but rather ...]", i.e. "Ich habe (nicht Geld)".
Thinking of sentences with "Ich habe nicht...", there is "Ich habe nicht die geringste Ahnung" (I haven't the slightest idea), but that probably falls under the "emphatic", similarly to "Ich habe nicht einen Hund". Or "Ich habe nicht die Absicht, dir weh zu tun" (I don't intend to hurt you; literally, I have not the intention to hurt you) which is probably the contrasty thing, parallel to "Ich habe nicht einen Hund, sondern eine Katze". Or the use of "haben" as an auxiliary verb, rather than a main verb -- e.g. "Ich habe nicht zugehört" (I didn't listen).