Prof Ponsati has been fighting the extradition charges in the Scottish courts.
In a tweet, Puigdemont said the drop of the warrants "demonstrates the huge weakness of the court case".
Ms Ponsati, who briefly served as Catalonia's education minister, had argued that the charges against her were politically-motivated, and claimed she would not receive a fair trial if she returned to Spain.
But it said that the Spanish charge of rebellion is not recognised in Germany and that related German statutes such as that against treason did not apply, because Mr Puigdemont's actions "did not rise to this kind of violence".
Mr Anwar stated that they are still awaiting official confirmation from Spain and cautioned that national warrants do remain in place.
Six other Catalan politicians are being held in custody on rebellion charges that carry a potential 30 year sentence. Under European law, that means Spain would have been barred from trying him on the more serious charge if the extradition were to proceed.
Puigdemont's Catalan government held an independence referendum past year, despite a Spanish court having ruled that the vote violated the constitution.
Judge Llarena wants Mr Puigdemont and his separatist allies who also fled the country to face charges of rebellion and sedition, as well as misuse of public funds.
Judge Llarena was scathing about the German court's decision, describing it as "a lack of commitment" in pursuing the fugitives.
They added that Spanish prosecutors have "refused to release Catalan political prisoners". But even in Spain, the rebellion charge had drawn controversy.
Military officers behind a 1981 attempted coup in Spain, for instance, were found guilty of rebellion.
Llarena counters there was, pointing to two events in particular.
Some police cars were damaged and the agents themselves blocked from leaving the building for hours.
The Spanish court rejected that proposal, lifting the arrest warrant altogether.