If you're the first person to post after the question has been sitting, idle and unresponded to for two days? 🙂
Seriously, though, "innovation" is, to my mind, something like "genius" in that it is very seldom identifiable at the moment, and often only identifiable in hindsight. "Innovation" and "genius" (which are not the same, but popularly thought about in similar ways) are both notions that are embedded in retrospection and historical perspective.
Detecting newness in doing requires awareness of long-term trends in doing. We can certainly identify when someone in a public sector proposes something "different" or surprising, but knowing when any such suggestion has changed the way things are done, or thought about, takes some time and perspective.
Is the degree of appreciation from others a reliable index of innovativeness? I'm not sure. Some innovations or bright ideas are noticed and appreciated instantly, some after a short adjustment period, and some not until years later. Some are noticed and resented, but in hindsight treated as positive direction shifts. Some are hailed immediately, but amount to a puff of smoke, or maybe even a bad or unwise direction.
Innovation, much like leadership, or genius for that matter, can exist at multiple levels. People can show leadership in grunt-level positions, at middle-management, senior management or even field-of-endeavour levels. It's all leadership although not all of it gets noticed (or compensated) in similar fashion. Similarly, innovation can happen at molecular or higher levels. It could be as simple as sorting the mail in the building or how admin support tasks are divided up. From that stance, innovation is a process or behaviour, rather than an outcome defined by impact. What one's mother does to whip up a spur of the moment meal for unexpected company from what's in the cupboard would certainly meet all criteria for "genius" as a cognitive process, even though the social impact is not quite that of a Thomas Edison, or Albert Einstein.
One needs to distinguish innovator from innovation. Few (if any) people are innovative 24/7. Some folks will have a terrific idea or two, and then pffft, rien, nada, garnicht, bupkes for years. Some will be bursting with ideas that go nowhere (here I'm reminded of actor Michael Keaton's character in the 80's film Night Shift, where he is constantly dictating what he feels are brilliant ideas into his hand recorder).
One also needs to distinguish between contexts for innovation. Some are, and need to be, much less tolerant of, or open to, innovation than others. And perhaps that's where the differences between private and public sector occur. As I was saying elsewhere here, part of what fosters the stakeholder trust that the public sector depends on to do its job is a certain level of consistency of functioning, rules, procedures, etc. In contrast, a company like Apple depends on always doing something new. I seriously doubt that anyone who flies a lot would appreciate quarterly changes in security protocol at airports.
In that sense, an "innovator" also needs to be sensitive to the tolerance for change, and appropriate rate of change within their category of organization.
There. Is that enough to chew on for a bit? Does it respond suitably to your questions?