I will concentrate on comparing (3) and (4). The definition (1) is meant for finite signed measures, whereas all the other definitions are meant for arbitrary positive measures; (1) is equivalent to (4) in the case of finite positive measures. (2) appears to be equivalent to (4) ["locally finite" can mean "finite on compact sets", although it is sometimes taken to mean "finite on the elements of some topological basis"; these are equivalent in the LCH (locally compact Hausdorff) case]. Finally, (5) does not appear to be a definition at all, but rather a description of a definition.
Now then,
i) In the case of a second countable LCH space, every locally finite measure satisfies both (3) and (4) (Theorem 7.8 of [1]). This is the most commonly considered scenario in applications, which is why almost no one bothers to carefully sort out the differences between the different definitions.
ii) In the case of a sigma-compact LCH space, (3) and (4) are equivalent. The forward direction is Corollary 7.6 of [1]; the backwards direction follows from the forward direction together with (iv) below (but I'm sure there is an easier proof).
iii) (3) and (4) are not equivalent in general, even for LCH metrizable spaces (Exercise 7.12 of [1]).
iv) In an LCH space, there is a bijection between
A) measures satisfying (3),
B) measures satisfying (4), and
C) positive linear functionals on the space of continuous functions with compact support.
(The Riesz representation theorem gives either (A)<->(C) or (B)<->(C), depending on where you look; (A)<->(B) is in the Schwarz book mentioned by Joe Lucke; see also Exercise 7.14 of [1])
[1] G. B. Folland, Real Analysis: Modern Techniques and Their Applications
Note: In [1], "Radon" refers to measures satisfying (3).
Well, we do know that $C_c(X)$ is dense in $L^1(X)$, so it's enough to check if $F$ is dense in $C_c(X)$.
Take $f\in C_c(X)$. This takes complex values in general, so we can write it as $f=f_1+if_2$ where $f_1,f_2$ are real-valued. It is not hard to see that $f_1,f_2\in C_c(X)$ as well. Now we can write $f_1=f_1^+-f_1^-$ and likewise $f_2=f_2^+-f_2^-$ and there are all positive functions with compact support. In other words, yes, $C_c(X)$ is indeed the span of positive functions with compact support. It suffices thus to approximate such a function through $F$.
Let $f\in C_c(X)_+$. Then we can write $f=|\sqrt{f}|^2$ so if we show that $\sqrt{f}\in C_c(X)$, we are done. But indeed, if $\sqrt{f}(x)\neq0$, then $f(x)\neq0$, so $\text{supp}(\sqrt{f})\subset\text{supp}(f)$ and closed subsets of compact sets are compact, thus $\sqrt{f}$ has compact support.
Best Answer
Yes, S2 holds without the assumption that $X$ be $\sigma$-compact. To see this, observe that the space of simple functions is dense in $L^1(\mu)$, so it suffices to show that simple functions can be approximated by functions in $C_c(X)$. Moreover, if $g \in L^1$, the set $\{x \in X \, : \, g(x) \neq 0 \}$ is $\sigma$-finite, so in fact it suffices to show that a simple function $\chi_A$ with $\mu (A) < \infty$ can be approximated by an element of $C_c(X)$.
Let $\epsilon > 0$. Since $\mu$ is Radon, for a $\mu$-finite set $A$ there exists a compact set $K$ and an open set $U$ such that $K \subset A \subset U$ and $\mu (U \setminus K) < \epsilon$. By Urysohn's lemma, one can choose an $f \in C_c(X)$ with $\chi_K \leq f \leq \chi_U$. It follows that $$\|f - \chi_A \|_{L^1} \leq \mu (U \setminus K) < \epsilon$$ as desired.