No matter how human a cyberlimb might look, to the piece of your soul that left with your flesh and blood arm it is the same as grafting a small backhoe shovel to your shoulder, or just leaving it off entirely. Hence the same essence cost.
I agree that the Essence affecting your Social limit is a change in
you, not the person you're talking to. When you lose a part of your soul, it makes you less human, and less able to understand and interact with other humans. It is very clearly stated in multiple editions that characters with very low Essence start to feel "detached" from their humanity and others, and even before they cross the line into cyber-zombie territory they risk disassociative disorders and cyber-psychosis. Sometimes there are rules for this, sometimes there aren't, but it's part of the setting.
That is why the effect on your Social limit is the same whether your essence loss is from internal cyber, bio, or chrome limbs, and whether you are talking to a mage, a sammie or an AI. It represents a disassociation with your own self.
Obvious cyberlimbs never get explained in much more detail than being "obvious". Ergo, it's natural everyone has their own opinion on how they should look, even the artists who make SR art themselves 
Personally, I think they just look like normal limbs in silhouette, but obviously (heh) mechanical somehow. Think John Kennox from Almost Human, or even the automail from Fullmetal Alchemist. Anything more substantial than that would most likely be a bulk-upgraded limb, and prove somewhat impractical.
A synthetic cyberlimb has a set appearance. It looks like an arm. Probably like
your arm, unless you bought it used, or got a sun tan since you picked it up.
An obvious limb doesn't have a set appearance. It is "obvious", which can range anywhere from a cheap model with skin-tone plastic shell and visible bolts and interface ports, a hyper-realistic arm that happens to be polished chrome, a steel and titanium frame with myomer wire bundles and pistons visible between the ceramic armor plates, or even a detailed work of art or engineering that doesn't even resemble the limb it is replacing. It can look like anything, as long as that appearance can be justified relative to the stats in the same.
If you have an arm that is AGI 10, it could be a smooth, precision-machined piece that looks like an impossibly lithe, chrome version of your real arm; or it could be a triple-jointed marvel of engineering that can move and bend in ways no metahuman limb ever could; or a hand-crafted work of art that twists and flows like a tentacle with a gleaming, multicolored surface of independently moving ceramic scales. As far as the actual effect in the game, all of these arms have the same cost and the same effectiveness.
It isn't about saying, "This is what obvious cyberlimbs look like." It is about saying, "I have this idea of what
my cyberlimb looks like, should I consider that Obvious or Synthetic?"