"I have not said that!"
Translation:Nie powiedziałam tego!
There is more important rule: you cannot separate "nie" from its verb. If you do, "nie" is applied to another word and it changes the meaning of the sentence. Here, it would be "Nie to powiedziałam." meaning "It wasn't this that I said.(I said something else)". "Nie powiedziałam tego." means "I didn't say this.(It's not important if I've said anything at all.)"
mówiłem is rather 'I was saying' powiedziałem is rather 'I said'
With the first verb you put the focus on the 'process of saying' something (i.e. I was saying good night, when I turned off the light.)
With the second one you put the focus on the fact, that you said something. (i.e. I said to him, that he looks great.)
Please correct me, if I misunderstood something.
Yes, but here 'that' is not a real determiner (it doesn't describe any noun), it's a dummy pronoun. This sentence could be "I have not said [that/this/it]!", all those versions would be okay in English, but it's still translated to "tego".
It actually works similarly as in a "This is Y" sentence, where all three: "[This/That/It] is..." would be translated as "To jest..."
In this sentence you're refering to a certain thing you (have) said. Nie powiedziałem tego - I have not said that. You have not said what? What exactly you have not said? I have not said that (tego). This is actually referring to Polish grammar rules and is called declination.
Dopełniacz (Genitive) - kogo? czego? (you may ask yourself an additonal question that should make it a little easier - whose have I not seen? Kogo/czego nie widziałem? (that question can be used for both "kogo" and "czego", and you might hear it when learning about declination)
"Kogo" is referring to people, "czego" is referring to things.
If you're really interested to read more, you can visit the website below (just don't spend too much time there as it might be overwhelming ;)
Cases. "to" is either Nominative ("To jest kaczka") or Accusative ("Powiedziałem to!"), "tego" is Genitive ("Nie powiedziałem tego!").
Genitive is used here, because if a sentence that normally needs Accusative is negated, it takes Genitive instead. (Other cases stay unchanged when negated).