Artificial Selection - Is the term simply convenient?

Well, perhaps the terminology is not what it should be, but the distinction is scientifically important. If you wanted different terminology you could call them intelligent and unintelligent selection. Now it is clear that intelligent selection can do things that unintelligent selection cannot. We can explain the legless dogs of your example on account of human whims; legless snakes cannot be explained by a whim of nature, but must be explained in terms of functionality. When we wish to test the theory that things in nature are the product of unintelligent ("natural") selection acting on variation, we are testing a stronger --- more precise and more demanding --- thesis than if we just asked whether it was all produced by some kind of selection, no matter what.

Well in that case we're not selecting which ones to kill and which ones not to kill in order with the intention of producing a new type.

Well the difference is that what you're doing in the first case, though it's certainly artificial, isn't selection. You're not picking and choosing among the sheep, or among the wolves.

Selection, in the biological sense, involves choosing among the members of a population, not between two whole species only distantly related.

Well, I'd have said that it was the differential reproductive success of the members of a population based on their traits. If this is because someone is picking and choosing which members survive and reproduce, that's artificial selection, if it just happens because some traits are more conducive to survival and reproduction, that's natural selection.