defamation, slander, libel, ...?
Solution 1:
According to the official (though not legally binding) English translation of the German law, "üble Nachrede" is (just) "defamation", while "Verleumdung" is rendered as "intentional defamation".
The equivalent article of Swiss criminal law on "üble Nachrede" is also titled (simply) "defamation" in English, while "Verleumdung" is translated as "wilful defamation".
This suggests that there are no established English terms to distinguish the two legal aspects as clearly as German does. The best option is to add clarifying adjectives, such as unintended vs. intentional.
(Austria, as of this writing, does not provide an official English translation of its penal code, but also distinguishes between the two in the original German.)
Solution 2:
In English CIVIL law, the difference between proven and unproven defamatory statements, whether slander or libel, is tested in the courts. If the supposedly defamed person sues the alleged slanderer or libeller in court, it is up to the defendant to prove their defamatory claim is true. The jury then decides the outcome, based on the evidence. If they find in favour of the defendant, the defamation is accepted as true. It is therefore expected that any publically declared defamatory statements should not be made unless there is evidence to support them.
One famous case is that of Deborah Lipstadt who was sued by David Irving for libel, by accusing him of distorting the facts in her book "Denying the Holocuast." The court ruled that Irving's claim of libel relating to English defamation law and Holocaust denial was not valid because Lipstadt's claim that he had deliberately distorted evidence had been shown to be substantially true.
The use of bad or offensive language, or "flipping someone off" is covered by other laws.