Should "State" be capitalized on its own?

I would say it's six of one and half-a-dozen of the other. First, here's the general pattern for AmE...

...and for BrE...

Those charts suggest that the modern trend (led by American usage, with Brits rapidly catching up) is not to capitalise.

Perhaps because I'm British and/or because I'm over 50, I tend to capitalise "the State" when I mean the British Government, Civil Service, NHS, etc., collectively, on the grounds that I think of it as proper noun referencing a single entity (the British State). But I don't capitalise forms such as state-sponsored industries, because it's a more "general-purpose" reference (to any nation-state).

I'm not going to wade through 46,700 (not case-sensitive) results in Google Books for passages containing BOTH "new york state" AND "the state pays", but I have to say my impression from glancing through the first few pages is that most of them are capitalised (in both terms).

I would say not, because in the second sentence it is no longer specified, but then I'm a mere Brit.