Why didn't Newton just propose the 2nd Law and leave it at that? Why didn't Newton just propose the 2nd Law ($F=\dot{p}$) and leave it at that? The 2nd Law implicitly contains the first, doesn't it? If so, it seems he wasn't following his own Rule #1 of Book 3 of his Principia: "We are to admit no more causes natural things than such as are both true and sufficient to explain their appearances." There are intro textbooks (e.g., this one) that study momentum first and then force $F=\dot{p}$ based on momentum. Cf. also this SE post and comments.
 A: Some of the problems regarding the first law are already pointed out and discussed in the question you linked to, so I'm not entirely sure where that part of your question comes from. 
I understand theoretical physics as the activity of coming up with mathematical models to give a structure to the behaviour of quantities, which the consensus agrees to be part of reality. One does this to thereby "understand" reality and to be able to calculate predictions (and also to make money and/or have some fun). From that perspective it's very possible to argue that the first law is redundant. The argument is that you have to declare an explicit form for the force for every model anyway, e.g. $\boldsymbol{F}_{ab}={q_1q_2\over4\pi\varepsilon_0}{\boldsymbol{\hat{e}_{a}}\over r^2}$, and at that point the first law truly becomes just a special case. In its history, there are other ways to formulate it or read new meaning into a first law but from my perspective they don't help much in doing physics in the sense above.
The third axiom is not already captured by the second, because at the very least, if you really only take space, time, at least two point particles and Newton's equations of motion into account, then nothing would hold you from investigating a mathematical structure with $\boldsymbol{F}_{ab}\ne -\boldsymbol{F}_{ba}$. This is then not Newtonian mechanics. Moreover, having that relation guaranteed also makes it possible to conclude general results about the theory for all the systems you might consider, e.g. conservation laws.
A: Newton wanted to sound like Aristotle. Aristotle gives physical laws in two parts--- there are the "natural tendencies" of objects, so that Earth wants to be at the center of the universe, fire wants to be on the sun, air wants to be in the middle, and water wants to be down, but above the Earth and below the air.
In order to make this happen, things go in straight lines at constant velocity. So Earth falls down, water falls down, air goes nowhere, and fire goes up. This is to make the natural tendency work.
But then there are external forces that make things deviate from their natural state. All of this is ridiculous nonsense, but this is what people called "physics" in Newton's time, and Newton had to deal.
When he is writing, he formulates the laws of motion to persuade the idiots, since he knows they won't read past page 3. So he says right up front:


*

*Hey, bozos, there is a new notion of "natural state". The "natural state" of all things is to move with a constant motion, no matter what it is. Sorry, I mean "all objects remain in motion forever unless disturbed".

*Hey, douchebags, the deviations from the natural state is by forces, which give you the time rate of change of the momentum. Sorry, I mean, "the impressed force is the magnitude of the deviation from the natural state".

*Oh yeah, dipshits,  look at this amazing new thing: the forces are equal and opposite! So that in any system, as much momentum leaves one body as enters another, and this is a universal law! Ahem, I mean, "every action has an equal and opposite reaction."


So now he has stated things Aristotle style (in Latin no less), and redefines the notion of "natural state" and "deviation from the natural state" so that first and foremost, it is correct, unlike Aristotle's bullcrap, and secondly, so that it is philosophically as compelling as Aristotle idiocies, if not more so.
Remember that Aristotle's physics had things that people considered laws of nature, like


*

*Nature abhors a vacuum: if you try to make empty space, things speed up to infinity to fill it.

*Nature doesn't make jumps: everything is continuous. No atoms.

*Nature proceeds in regularity: Everything moves in lines and circles


All this stuff is complete laughable nonsense, but philosophers thought it was deep. The "nature abhors a vacuum" meant that if you have a vacant position in the leadership, people clamor to fill it. Nature doesn't make jumps means you don't screw with tradition. And nature proceeds in regularity means you execute your radicals. Aristotle didn't care much about physics. He just wanted to justify the stupid politics of his time.
Anyway, Newton gives you new laws that have just as much political resonance:


*

*Everything keeps moving of its own accord: so nature keeps changing spontaneously.

*A force will change the velocity, not produce the velocity. This is resonant with the idea of "shifting the discourse slowly by political pressure".

*Nothing acts without being acted upon: there are no things that can do something without changing in the process of doing it.


These philosophical points, as trite and stupid as they are, are what gave Newton's proclamations weight with the philosophers. In this way, Newton beats Aristotle, and Newtonian mechanics (aside from being correct) is a suitable replacement for the Aristotelian garbage.
The philosophical nonsense is so dated, it is hard to take it seriously. Aside from the maxim "nothing acts without being acted upon" (which is a general law of universal validity, useful for other arguments in physics), the rest is just an embarassing demonstration that Newton was a politician as well as a physicist.
Today, we can dispense with the politics, and say "Newton's laws are the statement that momentum is conserved, and angular momentum too, if you assume that action at a distance between points occurs along the line of separation of the particles". Note that Newton didn't state "the world is made of particles that attract and repel along their lines of separation" (even though he believed this), he confined himself to macroscopic laws he was sure could be justified empirically without making hypotheses regarding microscopic constituents. In this regard, he was a true scientist. As regards the three laws, remember they are the content of the first 3 pages. Read the rest of the book, that's where all the interesting stuff is.
A: 
Why didn't Newton just propose the 2nd Law (F=p˙) and leave it at
  that? The 2nd Law implicitly contains the first, doesn't it? If so, it
  seems he wasn't following his own Rule #1 of Book 3 of his Principia:
  "We are to admit no more causes natural things than such as are both
  true and sufficient to explain their appearances." 

Simply because Newton didn't write the formula $F = m * a$. This was written some time in the middle of the XIX century.
If you want to know what Newton really wrote and the historical back ground and development you can find detailed information here

Newton thought that energy is linearly proportional to velocity.  The
  second law's original formulation reads: "Mutationem motus
  proportionalem esse vi motrici impressae" = "any change of motion
  (velocity) is proportional to the motive force impressed". 
This law, which nowadays is wrongly interpreted as: $F = ma$ (there
  is no reference to mass here) simply states states: $$[\delta] v
> \propto [\delta] Vis_ {motrix}$$ and in modern terms is sometimes
  (illegitimately) also interpreted as
  impulse,
  sort of : $$\delta v  \propto J [/m]$$. But mass is not at all
  mentioned in the second law (as the original text shows) but only in
  the second definition, where we can see a definition of momentum as
  'the measure of motion'

Quantitas motus est mensura ejusdem (motus) orta ex velocitate et quantite
    materiæ conjunctim = 'quantity of motion' (modern 'momentum') is the measure of the same (motion), originated conjunctly by velocity
    and 'quantity of matter' (total mass) 

and, moreover 'motive force' (vis motrix) is used, like all other
  scholars of the time, referring to the yet unknown kinetic 'force'
  that made bodies move, which Galileo had called 'impeto' and Leibniz
  'motive power' . The interpretation of this formula as the
  definition of force in modern usage is an ex post facto historical
  manipulation.
It was Gottfried Leibniz, as early as 1686, one year before
  the publication of the Principia, who first affirmed that kinetic
  energy is proportional to squared velocity (or that velocity is
  proportional to the square root of energy): $$ v \propto
> \sqrt{V_{viva}} [/m]$$. He called it, a few years later, vis viva =
  'a-live' force in contrast with vis mortua = 'dead' force:
  (Cartesian) momentum (mass/weight * speed: $m *|v|$). This was
  accompanied by a first formulation of the principle of conservation
  of kinetic energy, as he noticed that in many mechanical systems of
  several masses $m_i$ each with velocity $v_i$,
$\sum_{i} m_i v_i^2$
was conserved so long as the masses did not interact. The principle
  represents an accurate statement of the approximate conservation of
  kinetic energy in situations where there is no friction or in
  elastic collisions. Many physicists at that time held that the
  conservation of momentum, which holds even in systems with friction,
  as defined by the momentum: $\,\!\sum_{i} m_i v_i$ was the conserved
  kinetic energy........
  Full post at the quoted link.

As to the rule you quote it is not his rule but the centuries old Law of parsimony better known as Occam's razor

The words attributed to Ockham, entia non sunt multiplicanda praeter
  necessitatem (entities must not be multiplied beyond necessity), are
  absent in his extant works;[21] this particular phrasing owes more to
  John Punch.[22] Indeed, Ockham's contribution seems to be to restrict
  the operation of this principle in matters pertaining to miracles and
  God's power: so, in the Eucharist, a plurality of miracles is
  possible, simply because it pleases God.[17] This principle is
  sometimes phrased as pluralitas non est ponenda sine necessitate
  ("plurality should not be posited without necessity").[23] In his
  Summa Totius Logicae, i. 12, Ockham cites the principle of economy,
  Frustra fit per plura quod potest fieri per pauciora [It is futile to
  do with more things that which can be done with fewer].

His versiom became very popular but the princple dates back to Aristotle (ibidem)

Aristotle writes in his Posterior Analytics, "we may assume the
  superiority ceteris paribus [all things being equal] of the
  demonstration which derives from fewer postulates or hypotheses

In the same wiki article you'll see that it is confirmed that Newton's rule was not original
