It makes absolutely no sense whatsoever for the other green (the one without the flagstick) to NOT be declared a wrong green and preventing players from taking relief from it.
Why would you want to let players hack up a painstakingly prepared putting surface with a wedge or an iron, where that green is typically nowhere near the actual green in play? To put that green in play the following week or whatever? Except, now you can't because it's all damaged!
The rules of golf may seem complicated and what not, but they are derived from common sense (and fair play), and I agree with Rulesman's interpretation that if indeed any putting surface was the considered to be "the green", the term "any" would have been used.
Now, if the 2 "greens" are in fact contiguous to each other with perhaps a strip of rough between them in one area (but the contiguity is maintained in another part), then it's a single green, and it would be no different than having a bunker in the middle of the green: it's not necessarily a smart or cost-effective design, but it's possible, even when only amateurs and hackers are using that course. There is such a hole at #15 at The Ranch, in San Jose, CA.