And all of them are ancient (different time, different culture, very different industry) and not one of them is an action blockbuster; not particularly relevant comparisons since their impacts would be completely different than a film like Avatar. Films are more disposable entertainment these days; you look back at not just Psycho, but all of Hitchcock's films, and they're all classics, many took a while to gain cultural traction (Vertigo not until its remaster almost 3 decades later); but that's not particularly true about the last 15 years or so - oversaturation? The Internet? Who knows the reason. But usually what makes a film culturally relevant among new franchises is when the sequels or TV series comes out.
How about Inception. Released a year after Avatar, and based on the amount of times it's been ripped off, the amount of times it's referenced in jokes (mostly playing on the word Inception), the amount of times I've heard it's music or sound-effects used in youtube videos, memes etc., I'm willing to say it's had more of a cultural impact than Avatar. That's just one off the top of my head.