How Did Newton's Second Law Get Its Definition?

Newton didn't say "change in momentum", he said "alteration in momentum", and whichever he said, this means clearly, and with no room for doubt, rate of change of momentum, the limit of small $\Delta t$ of $\Delta P \over \Delta t$. This was understood this way by everyone who read the book, there is no way to misinterpret if you follow the mathematical things.

The experiments of Galileo showed that bodies in gravity have the same acceleration. This means that the Earth is imparting changes in velocity to particles. The notion of "force" is already present to some extent in the theory of statics developed by Archimedes, and gravity produces a steady force in a static situtation, and this force is proportional to the mass. If you know force is proportional to the mass, and the acceleration is the same for all bodies, it is no leap to conclude that a force produces a steady acceleration inversely proportional to the mass.

The second law was not the major innovation in Newton, this was known to Hooke and Halley and Huygens for sure. Newton's innovation is the third law, and the system of the world, the special problems.

Isaac Newton, Philosophiae Naturalis Principia Mathematica, Axiomata, sive Leges Motus [1]:

Lex. II.
Mutationem motus proportionalem esse vi motrici impressæ, 
& fieri secundum lineam rectam qua vis illa imprimitur.
[...]

Mutationem: means change, alteration, variation, mutation, ... Motus: momentum.

So Mutationem motus means exactly the variation of momentum: $$ \frac{\Delta p}{\Delta t} $$

[1] http://la.wikisource.org/wiki/Philosophiae_Naturalis_Principia_Mathematica