what burden of proof
To defend against a libel lawsuit, probably a "preponderance of evidence", what ever that means in the particular jurisdiction concerned.
Certainly not "proof beyond reasonable doubt" (as standard that only police insist applies to their employment contracts).
Certainly not "a balance of probabilities".
To defend against a wrongful dismissal lawsuit, that depends not only on the jurisdiction but also the employment contract, either "balance of probabilities" "preponderance of evidence".
I'm not a lawyer, but I think cases where the employer has done a press release explaining the dismissal as being for sexual harassment, in the USA and UK the potential damage from a libel lawsuit tends to dwarf the risk from a wrongful dismissal lawsuit.
Basically that press release is going to hurt chances for ever being trusted by a 'first world' employer again. I expect future jobs will come with close supervision.