Inflammations are non specific defense reactions instigated on cell injuries.
Inflammation reactions are characterised by certian physical changes that are inturn caused by physiological changes.
The physical changes are:
- loss of function
Here the redness is caused due to vasodilation of the nearby blood vessels (mostly capillaries) which means that the amount of blood passing through the vessel is increased and thus the region appears red. But this doesn't end there, as the vessels dilate their permeability to fluid and leukocytes increase that ooze out into the tissue and cause the region to swell up.
The resident phagocytes (phagocytes present in body tissues including skin) together with these leukocytes fight against the invading microbes. However microbes that are present all around us, even abundantly on skin get a chance to enter our body through the smallest cut or tissue damage.You can say inflammation is a defense mechanism that the body employs to mitigate any imminent infection.
Now coming to your questions:
Those injuries aren't an infection but could these things still be called an inflammation?
Yes they are inflammations. If they are displaying the classical symptoms then they are inflammations.
Or with other words, is there always an inflammation when cells are damaged or is needed more?
Yes, there's always an inflammation if cells are damaged.
Is it necessary that in both cases cells are damaged to induce a cascade of hormones?
No in none of the cases mentioned by you a cascade of hormones is induced. Rather in both cases; INFECTION and INFLAMMATION a number of immune cells are induced to carry out a cascade of reaction which actually identify the microbes and cause
However an infection can be caused without cell injury, e.g. pathogens entering through nose and mouth.
Note: Phagocytosis etc are complex immune reactions that occur against pathogens and there are chapters dedicated to them in Immunology books . I would suggest you to go through a google search to get a general idea if you are curious enough to know what they are.