So, what makes those No Man's Land "Batcaves", Batcaves other than name? Isn't a Batcave Batman's hideaway located beneath Wayne Manor? Not all of them have stalagmites, just look at the pic you posted. Some of them are actual foundations, just like the bunkers. Isn't one of them even located at a Wayne Industries center? Or does white decor make it a bunker? Does a character have to refer to it as "Bat Bunker" for it to be legit? Bruce and Alfred never call it that in Dark Knight, the only thing that is spoken is "Batpod".
You mention structures, and I agree with you. But those headquarters from No Man's Land were all different and had varying looks. Some were Wayne Enterprises, some were within the city, hell, one was even at Arkham (unless I'm mistaken and am getting mixed up with the video game). They all served the same purpose that TDK and current comics had. Temporary headquarters.
Then if you really want to get specific, yeah, the comic bunker has a sort of whitish decor and a sterile look (but then again, it doesn't have those over head lights the film has and is mostly greyish), but other that and name what else is there?
So the connection to the Nolan creation is name and I guess look. If we want to get technical here, where were the multiple levels of Nolan's bat bunker? The latter we see Alfred coming down from? The subway station (with tracks that lead to the Batcave)? In the film, it wasn't located under Wayne Enterprises or even Wayne's penthouse, it was on the outskirts of Gotham by the docks on some other Wayne industries property. All it was a fortified area with hidden panels.
This looks a lot different,
Basically, my point is, Batman has had other establishments throughout the city other than THE Batcave before 2008 and Nolan. Doesn't matter what they were called. My point is that, everyone is inspired from one another and it's an involving thing. I don't care if it's a game, a comic or a movie. They all take and give to each other creating the Batman we know currently (which is constantly changing). Black Batman armor didn't always exist. Either did a suit vault. Neither did the term "bat bunker".
I'm in complete agreement that the comic borrowed/was inspired by 2008, but it's not like that idea of "Batman operating out of a place other than the Batcave" is new or original. Those previous pads of Batman WERE bunkers. All temporary fixes for him since they weren't THE Batcave.
Can I find you a pre 2008 Batcave that has white lights and is called a Bat-bunker? No. The Batcave has taken on many different looks, from dank and cluttered to modern and sterile. Sometimes there's a T-Rex in there, other times, it's nothing but sleek, flat, clean architecture. Sometimes there's an entrance via clock, sometimes book case.
All I can say is, those No Man's Land are essentially "bat bunkers", in function, even though they're not white and not called "bat bunkers" directly. In fact, I'm not even sure if they're called "Batcaves" either.
And yes, TDK/Nolan did inspire the bunker in #687, the name anyway but there have been previous hideaways just like it previously. Even if it wasn't white or whatever.