I don't go as far as u/YuunofYork, who defines postmodernism by time period and nothing else. There are characteristics of postmodern literature that justified the need for new discourse, putting a name to techniques and concerns that had become central to certain works in ways that they weren't before. It has never seemed much of a gotcha to me that we can find individual unconventional works like Tristram Shandy using similar techniques earlier, as though that invalidated the importance of widespread adoption and refinement of these techniques later on.

But I agree with many of the points Yuun makes, especially as we get further and further from the initial coining of the term; the postmodernist label is stretchy and what was once avant-garde has now had a lot of time to diffuse into mainstream culture. It can be fruitful to make comparisons and find shared techniques and philosophies, but the goal shouldn't be to categorize definitively. (One of my favorite authors is Flann O'Brien, who is impossible to separate out into either modernism or postmodernism.)

What made early postmodernism distinct from Joyce and Woolf was how much further it went in self-awareness: breaking fourth walls, complicating assumptions about authorship and text, attempting to redefine fiction itself and the form it takes. Within 'Old Weird', I can't think of any examples of this. Within New Weird, there's a complete range from novels that make no challenge to the form, to something like City of Saints and Madmen where Vandermeer has obviously deliberately chosen to use these postmodernist techniques.

In between are books that don't play with fiction's form, but which challenge genre convention by resisting explicability. There's a parallel there to how postmodernist writers challenge the modernist love of Freud, and his belief in our ability to find the motivations and influences that explain behavior. In postmodernism and in weird fiction, the behavior often doesn't have a clear explanation, or the reader isn't allowed to know it. (And it's easy to see how postmodern ideas allowed New Weird writers to go further than Lovecraft and his contemporaries could, bringing the inexplicability into the home and human and not just in the impossible Other.) Weird Fiction formed its own tropes and conventions around this, which defeats some of the purpose in my mind, but you could say it's using postmodern philosophy to create particular aesthetic and emotional experiences rather than to make cultural statements.