It's not just some thing that happens. People realize they can make more money, so they do. Happens at every step in the supply chain, and thus prices go up across the supply chain.
Prices are set at the limit of what the market will pay
Oh no. Not for essentials and definitely not in our current economic environment. Our economy isn't designed to have people spend money only on essentials. When people only spend money on rent and food or whole economy is on danger of collapsing.
No... Well not in a detailed way and ignores greed. So which would your definition go under demand-pull inflation, cost-push inflation, or built-in inflation.
Greed is always a factor though, this isn't a new variable to account for. All three categories you provided have a profit motive as the core factor, it's simply in different layers.
A real example of inflation that isn't tied to an individuals desire to maximise their benefits from a transaction would be something like extracting an ore than becomes harder as the rich deposits are used up.