Fleming always wrote it as the Double-O Section.
It's also usually referred to that way in the films, and often as 'the department' too (e.g. In OHMSS, M tells Bond "This department is not concerned with your personal problems"). In Skyfall, when M is appearing at the Enquiry, she says "Today I've repeatedly heard how irrelevant my department has become. Why do we need agents? The Double-O Section?" So we can safely rule out that it was part of the reboot.
I'm not sure why it was changed in Spectre, but it's interesting that's M, Bill Tanner and Max Denbigh, all refer to it a single the Double-O Programme at various points.
MI6's own website refers to 'departments' rather than programmes or sections, so I can't believe it was written that way for accuracy.
The original Spectre script was written by John Logan, but later Purvis and Wade were hired rewrite or polish it. Why they didn't change it back - assuming Logan wrote it that way by accident - is a mystery.
"How was your lamb?" "Skewered. One sympathises."