If they walk their walk and talk their talk, that's what matters most. That to me is integrity. They are sincere and have no hidden agenda.

The notion of one's actions being in line with their speech, and being consistent across all aspects of one's life makes sense, which leaves out concerns over what one believes. This is okay for most cases, but can I call and true and fast KKK member integrous? No. Because to do so would run counter to my own sense of integrity.
So maybe the word has both a literal, dictionary definition and also an implied one.

My definition of integrity would be having principles and a strong moral code and not allowing yourself to be persuaded by others, by bribery or reward or coercion to set them aside.

its a huge part of what led to my divorce. The same person who was using creams, talking about eating healthy,etc, was hiding alcoholism for over fifteen years. Then she started smoking after quitting for 25. She is not always trying to lie, but lies by omission often. She told me recently she stopped drinking beer. My son had already told me it was mostly wine. Integrity matters and to me its about making sure your actions line up with what you say.

I'd love to wade in on this one, but i'm just too tired. My short story is therefore that I would perceive someone with integrity as having demonstrated honesty and sincerity.

That's really too narrow a description when i think about it, but is all i can articulate as my fingers want to go to sleep, and now. hahaha

To me, integrity is doing the right thing even if no one is looking.