The name pseudomembranous is more of histology. It is a false membrane/ exudate overlying the mucosa.

I agree. For example, Clostridium difficile infection is the most well defined example of pseudomembraneous colitis. When the initial infection begans, plaques form on the surface of the colonic mucosa. This is due to the infiltration of polymorphonuclear cells in response to the infection that subsequently damages the epithelial lining. A fibrinous exudate results that will eventually coalesce to form a "pseudomembrane" and typically does not result in ulceration of the underlying tissue.

As in pseudomembraneous colitis, not genuine, but pseudomembranes are formed from infalmmatory exsudates.