Okay, what is N1, as distinct from a special case of N2?
Good question.
The answer is that N1 is by no means a special case of N2.
1) For instance, you could imagine a world where the acceleration depended
on force in some non-Newtonian. This doesn't even take much
imagination; in a relativistic world F=ma is _not_ the right answer.
Meanwhile, N1 is tantamount to a statement of conservation of
momentum. That remains true, relativistically or otherwise.
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2) The complementary counterexample can be imagined also, with a little
more difficulty. Imagine a "skyhook" that comes down from nowhere and
applies a force to the side of my test particle. By recording the
position, and using a strain gauge on the side of my test particle, I can
observe that F=ma, but the force has no equal-and-opposite counterpart and
momentum is not conserved.
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If that doesn't answer the question, please clarify the question.