It doesn't matter where Edwards got the idea from...all it is is an Alien Invasion movie. That's it. It's Independence day set in a Mexican forest for low budget. The only differences: Smaller scope POV, Aliens were different, we didn't see them till the end and it was filmed on a prosumer cam.
Signs is Independence day set in a farmhouse.
That Cloverfield One with John Goodman is Independence Day set in a basement.
Independence Day itself was just War of the Worlds again. With added jingoism, airplanes and less theme.
You choose what to show. A big budget film can show military from all over the world attacking huge space fleets, a low budget film may have the same action, but instead two people listen to the attack being reported on a CB radio.
You also get to choose what else the story is about...from theme, to genre beats, twists etc. all of which give it a different feel, if you desire.
Ultimately, the difference here seems to be purely one in interpretation:
I get what they mean. They are not asking for the LITERAL film they mention to be made low budget, they are looking for things with High concept, big emotional bang in that genre that is attainable on a small budget. There are no limits on creativity, just on scope. The moment in Signs when they see that singular alien on the old TV set was great (despite poor VFX)...it was a more memorable moment than anything in Independence Day...that's what they're after. Big impact, small scope.
That's where the creativity lies...doing something we've seen lots of times, but in a slightly new and affecting way.
If a Producer asks for something "in the vein" of something else, they're not demanding that you follow the exact template.
What they're asking for is creative, high concept, strong genre ideas that can be filmed well and convincingly on a small budget. They are not asking for generic scripts that can't be filmed on a small budget...like the Rim of the World example.
Your last line is dealing with two different situations:
The Producers putting the shout out are usually looking for scripts that are already written...so are using existing IP's to give the writer a sense of whether their story fits into what they are looking for. They give a few examples so as not to suggest they are looking only for an extremely specific type of story.
Your example deals more with being assigned to write a film. That aside:
"I’d much prefer the creative challenge of, What if aliens did invade Earth 10,000 years ago and it was us humans? GO!"
Is just an aspect of Oblivion. You want to write Oblivion on a micro budget, basically....which is fine, but it's the same thing as:
"Rather than, a movie like Minority Report but on a micro budget, GO!"
It's just that you're failing to deconstruct it into: What if it was possible for future Police to predict crime before it happened? What effect would that have on society?
Or what if it were possible to judge human beings on their thoughts? Etc.
Or How could I commit a crime in a world where the Police can predict your actions and can read your thoughts?....And in writing that it occurs to me someone already made that film...they answered that by having him learn Buddhist meditation to clear his mind, so he could destroy the program he'd invented that was being used to monitor everybody.
https://www.imdb.com/title/tt3153582/So, you see, Minority Report on a Micro Budget is exactly the type of thing writers and directors should aim for. You've just got to learn how to ask the proper questions and break it into its constituent pieces.
It's all the same thing in the end. Like an actor interpreting directions, the writer has to learn to interpret what the Producer is actually asking for.
As I say, they're not looking for generic refits, but creative works that could change their lives that are nevertheless attainable on tiny budgets.
Which they mostly won't find, of course, which is why most Producers/Directors that make it end up making their own stuff..like Edwards.
That works both ways, of course...the writer who is sick of filmmakers messing their stuff up, and the Director who can't find the right story...they both end up in the same place.