PF: Does a three-month visit to another country count as “working outside of their country”, so that an applicant will be eligible to apply for the 2022 PF call in their country?

In principle, yes, if for the 3 months the main activity would be in the other country, they would be eligible. However, if for example this is a secondment and their employment contract remains with the organisation in their home country, it probably does not classify as such and would rather be a ’short visit’ in the sense of the mobility rule.