I think I understand the dilemma here. Every idea in physics has two sides: the mathematical abstraction and the physical realization e.g. we have some intuitive notion of velocity in our heads through the physical world which can be accurately modeled and abstracted as some vector in Euclidean space (in classical mechanics at least). You want the latter and I think a better description of the former.
An event's mathematical abstraction is a point in <PERSON> space-time (don't worry, I'll get to the physical realization in a moment). An important clarification here is we are not working in a vector space, per se. We are working in an affine space i.e. one with no origin (intuitively, this makes sense, space has no origin). This means there are not unique coordinates for any individual event (even in the same inertial frame)! The only concept that really makes physical sense is a vector between points not the vector of the points themselves. Lorentz transformations work only on space-time vectors and hence only between these differences of points i.e. $\Delta x, \Delta t$, etc. If you want to read more on this notion of affine spaces in the context of classical mechanics, I refer you to <PERSON>'s beautiful book on mechanics. This leads us to change our train of thought from singular isolated events to pairs of events because only those have any relation to <PERSON> transformations.
We know, from basic vector calculus, that vectors can be written in many different basis. The same applies to these <PERSON> space-time vectors. Each inertial frame is simply a different basis. Sounds cool but doesn’t make quite that much sense yet, I know. It will become clear in a second when we talk about what an event really is physically. 844
<PERSON> transformations, are simply a basis transformation i.e. physically a frame change. Now to the moment we have all been waiting for.
The physical realization. We have said above that we must think in pairs of events. Lorentz transformations help us see the vector between these two events (i.e. the spatial and temporal difference) in different frames. Now, we simply define an event as something that happens.
For example, a train entering and exiting a tunnel. This is an event. There is a frame-independent way of seeing if the train’s front is at the entrance and the train’s front is at the exit. Simply when they are aligned. Great! Now <PERSON> transformations tell us that if we take the spatial and temporal difference between these two events in one frame, we can use Lorentz transformations to figure it out in a different inertial frame. Yay!
Note that it is important that those events were frame independent. For length contraction, the two events are the measuring the position of the front of the stick and the back of the stick at the same time. However, in different frames these are actually different events because our measurement was frame-dependent i.e. we forced simultaneity.
I hope I answered your question explaining how we should think about events and their coordinates, what the <PERSON> transformations mean physically and mathematically, and what events are physically and mathematically. Let me know if there is anything I missed! 844