Natural immunity is what anyone who gets vaccinated or infected acquires, assuming that the immune response is successful.
Immunity acquired from infection is more regularly referred to as convalescent or infection-induced immunity; moreover the messaging has consistently been that while there have been some studies that convalescent immunity can potentially be as or even more effective than vaccine induced immunity, it comes at a significant increase of complications, ranging from organ damage to the fairly nebulously defined long covid, and most of the studies that are credible in the field that have endorsed the effectiveness of convalescent immunity still recommend vaccination to avoid the potential impacts of even "mild" infections.
The rampant claims about so-called natural immunity by people may be drowning out correct or rational voices, but that shouldn't indicate that the messaging has changed, just that the messengers are spreading misinformation.