There are numerous ways of dealing with Newton's Laws. The ubiquitous introductory text by Halliday and Resnick, Fundamentals of Physics, treats the First Law as a special case of the Second Law in which the net external force is zero, just as you have done. Goldstein, in Classical Dynamics, also gives the Second Law a privileged position.
On the other hand, the widely-used text by Marion and Thornton, Classical Dynamics of Particles and Systems, treats the First and Second Laws as definitions, and the Third Law as the actual physical law.
However, Marion and Thornton also refer to Lindsay and Morgenau's book Foundations of Physics, which they say presents the First and Second Laws as physical laws and the Third Law as a consequence. (I have not looked up this reference, so I don't know how they do it).
Ernst Mach, in his Science of Mechanics, condensed all three laws into one law, which makes no mention of either force or mass. His version goes like this: "When two compact objects act on each other, they accelerate in opposite directions, and the ratio of their accelerations is always the same." It is easy to see that working definitions of force and mass can be derived from this.
Newton's Laws were formulated three centuries ago, and they have been stated in many different ways since then. Which statement is best is clearly a matter of opinion. However they are stated, they seem to provide a good way to deal with problems of non-relativistic motion (although not, by any means, the only way; for certain problems, Lagrangian or Hamiltonian formulations are simpler). Since we are engineers, this is what we really care about: a satisfactory set of equations for solving practical problems.