So, here comes my question: Are all of the technical specs, like output voltage, channel separation, frequency response the domain only of the stylus itself or does the cartridge itself influence it in any way?
The stylus traces the record groove. Period. It definitely impacts the sound, but it is far from being as decisive or important as you seem to think.
Stylus shape determines how it fits in the groove. The goal is to follow the groove wall. Why? As the stylus moves left and right, up and down, what is it doing?
The stylus is at the tip of the cantilever. That's the thin little wire you see coming out the bottom of the cartridge. As the stylus moves back and forth it wiggles the cantilever.
Why is it called the cantilever? Because its a lever. The cantilever goes way up inside the cartridge where you can't see. There's lots of different designs but basically they all look something like this:
http://turntablebasics.com/images/ATMagnetics.jpg
This is what's called a moving magnet (MM) cartridge. Moving coil (MC) is the same only the magnets and coils are reversed. That's not the point.
The point is the stylus traces the groove, which moves the cantilever, which moves the magnets (or coils) which makes the signal.
Every single one of these component parts has a huge impact on the sound. Its not just the stylus. Its all of them.
Frankly: can any other cartridge than the original one have impact on the sound? Does cartridge body itself has any properties when it comes about sound quality and characteristics?
Oh good, you asked about the cartridge body. Yes that too.
I understand when you said "cartridge body" you lumped all the inside stuff in the picture above together with the outside of the body you can see. When in fact the body is one more component part of the cartridge.
What happens is every time you play a record the stylus starts moving back and forth which causes the cantilever to vibrate which causes the whole suspension and magnets and yes the cartridge body to start vibrating.
The problem is the signal we wind up with coming out the cartridge is the result only of the magnets and coils moving relative to each other. If you look at the way this is all made you can see there is no way of separating the stylus vibrations that are the signal we want from all the other stuff vibrating that we don't want.
So yes even the cartridge body itself, by the way it helps (or doesn't) to control all this vibration has a huge influence on the sound.
That's what makes these things so expensive and hard to make. Every single one of these tiny little parts affects the sound. The stylus is the least of it. Well, actually not any more the least than any of the other parts. They all matter.
The beauty of this is you can pretty much forget about the stylus. You can forget about the cantilever. All you need to think about is how much voltage does it put out, and how does it sound? You want enough output for your phono stage. That's first. Once you have that then you want it to sound as good as possible for the money. That's it. Whatever stylus or body or whatever has enough output and sounds good enough, that's the one.