What is an event in Special Relativity?

In the original usage, as Einstein used it, an "event" is just something that happens, like a detector clicking. It's just the same as the colloquial meaning.

In the early 20th century, various thought experiments involving hypothetical events, and actual experiments involving physical events, were used to show that general relativity is an excellent model for our universe. In the context of general relativity, spacetime is modeled as a Lorentzian manifold, and physical events are modeled as points in this manifold.

Now, some mathematically minded people choose to forget all this history. They instead say the word "event" is defined as a point in a Lorentzian manifold. This is a clean and consistent definition, but as usual in mathematical physics, it misses the point. The only reason we care about these mathematical "events" is because they form part of a theory that does an excellent job of describing real events. By conflating the two, one glosses over the mountains of experimental work needed to link the two together.

So you are completely right to notice that something fishy is going on. This linguistic bait and switch occurs constantly in physics courses. Typically one starts a course by defining spacetime and event in the usual, colloquial sense, but then ends the course by saying that spacetime is a Lorentzian manifold, that an event is a point. These are extremely different meanings of the same word, which are both commonly used, and it's important not to conflate them. The gap between them can only be bridged by experiment.

An event is any physical occurrence that we can consider to happen at a definite point in space and at a definite instant in time. These could be approximated to the location of the nearest intersection of imaginary grid points in space, each carrying a synchronized clock. There is a famous picture of such a reference frame in the book by Taylor, Edwin F., and John Archibald Wheeler. Spacetime physics. Macmillan, 1992.

The numerical values $(x,y,z,t)$ uniquely locating an event depend on a certain number of arbitrary choices, such as the location of the origin of the lattice and the moment considered to be $t=0$.

Event can refer to real events which are for example: - a firecracker exploding - two objects scattering - a photon getting emmited from an atom

Events also.dont have to.be real, it is enough (of course) that you could imagine e.g. a firecracker beiing exploded.

More abstractly an event is a point in space and time to which you can refer to.without using coordinates in space and time.