As I said, my installed version is newer than the download page lists.
So, i tried that download, anyway. I get:
This update requires OS X version 10.9.5.
If you know that you have the same version as the installer suggests, then you are good to go.
I think that your installer may be written to respond like that, because the firmware may be updated through other methods, such as a macOS installer.
My mini firmware has updated at least twice, maybe three times (not sure), since the release of that manual updater.
I expect that Apple does not provide an installer for those later updates, because the firmware is updated during a macOS update, with no separate release made available. It's a one-time update, when the macOS system demands it during the macOS update.
Does that make sense?
I do know that your 2010 MBPro was updated specifically to allow internet recovery to work on that model. If you still have the same firmware as was released in that 2012 update, then nothing newer has been required for your particular model, and apparently no updates released with macOS installers, either.
Just curious - what version of OS X are you running on your MBPro?