In H. Halbertsam and H. Richert's book "Sieve Methods" (Academic Press 1974) the authors go about illustrating the proof of Chen's theorem in Chapter 11 (Pages 321-337). They use a function $S_0$ defined on line (2.4) at the bottom of page 323. I simply want to know how this $S_0$ relates to the number of Chen Primes less than a given magnitude $N$. Are they the same?
No, they are different. This sum $S_0$ is just the contribution of numbers with three prime factors to the counting function used by Chen.
Roughly speaking, Chen first proves a lower bound for a weighted sum (say $S$, which may be different notation from Halberstam-Richert) over primes $p\leq X$ such that $p-2$ has at most three prime factors. He then gives an upper bound, which turns out to be of smaller size, to the contribution of the integers with three prime factors, and deduces that the contribution of those $p$ where $p-2$ has at most two (the "Chen primes") must be large, hence proving his theorem.
This idea is used frequently in the deepest applications of sieve and is often called "switching primes" or "switching trick" (see the references in Friedlander-Iwaniec "Opera de Cribro", the term is in the index, and it is used in their version of Chen's Theorem in Section 25.6).