$\newcommand{\H}{\mathrm{H}} \newcommand{\Z}{\mathbf{Z}}$Let $X$ be a space. Then the $E$-(co)homology of $X$ is the same as the $E$-(co)homology of its suspension spectrum, i.e., $E_\ast(X) \cong E_\ast(\Sigma^\infty_+ X)$ (and you remove the basepoint in $\Sigma^\infty_+ X$ to get reduced $E$-homology). In the case when $E = \H\Z$, this says that $\H_\ast(X;\Z) \cong \H_\ast(\Sigma^\infty_+ X;\Z)$. In the case when $E$ is the sphere spectrum, this says that the stable homotopy groups of $X$, i.e., the unstable homotopy groups of $QX$, are the same as the homotopy groups of the spectrum $\Sigma^\infty X$.
Let's now turn to the homology of $QX$. We know that $\H_\ast(QX;\Z) \cong \H_\ast(\Sigma^\infty_+ QX;\Z)$, so you need to understand $\Sigma^\infty_+ QX$. A theorem of Snaith's says that there is an equivalence
$$\Sigma^\infty_+ QX \simeq \bigvee_{n\geq 0} (\Sigma^\infty X)^{\wedge n}_{h\Sigma_n},$$
so we find that
$$\H_\ast(QX;\Z)\cong \bigvee_{n\geq 0} \H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\Z).$$
The groups $\H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\Z)$ can be computed by a homotopy orbits spectral sequence:
$$E_2^{s,t} = \H_\ast(\Sigma_n; \H_\ast((\Sigma^\infty X)^{\wedge n};\Z)) \Rightarrow \H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\Z).$$
If $\Z$ is replaced by $\mathbf{Q}$, then this spectral sequence degenerates (the $E_2$-page vanishes for $s>0$), and you find that $\H_\ast((\Sigma^\infty X)^{\wedge n}_{h\Sigma_n};\mathbf{Q}) \cong \H_\ast(\Sigma^\infty X;\mathbf{Q})^{\otimes n}_{\Sigma_n}$, i.e., $\H_\ast(QX;\mathbf{Q}) \cong \mathrm{Sym}^\ast \H_\ast(\Sigma^\infty X;\mathbf{Q})$. By the way, something special happens with $\mathbf{Q}$-cohomology of spectra: since the $\mathbf{Q}$-localization of the sphere spectrum is $\H\mathbf{Q}$, you find that $\pi_\ast(F_\mathbf{Q}) \cong \H_\ast(F_\mathbf{Q};\mathbf{Q}) \cong \H_\ast(F;\mathbf{Q})$ for any spectrum. (This is more generally true for any smashing localization.) (Edit: Whoops, I didn't actually finish my answer.) This means that $\H_\ast(\Sigma^\infty X;\mathbf{Q}) \cong \pi_\ast(\Sigma^\infty X_\mathbf{Q}) \cong \pi_\ast(\Sigma^\infty X) \otimes \mathbf{Q} = \pi_\ast^s(X) \otimes \mathbf{Q}$, so you find that
$$\H_\ast(QX;\mathbf{Q}) \cong \mathrm{Sym}^\ast(\pi_\ast^s(X) \otimes \mathbf{Q}),$$
as you said in your post. The homology of $QX$ with coefficients in a general homology theory $E$ is a lot more complicated: if $E$ is a structured ring spectrum, it describes what are known as Dyer-Lashof operations for $E$.
The details of the comparison are treated in detail in the original ABGHR paper (and then unfortunately split in half across two papers in the updated version), so I'll just try to give a sketch of what's going on.
Thom spaces
Given any type of bundle $E \to B$ we can view this as a diagram of spaces indexed by $B$. The homotopy colimit of this diagram in the homotopy theory of spaces is just $E$ itself (which makes sense- we are 'summing up over $B$').
If $E \to B$ has a section $\infty: B \to E$, then this supplies each fiber with a basepoint. The homotopy colimit over $B$ of this diagram of pointed spaces can't be $E$, because it has a whole $B$-family of basepoints. So we need to identify those all together and take the cofiber $B \to E \to E/B$. The homotopy colimit is then $E/B$.
Example. Given a pointed spherical fibration, $E \to B$, with fibers $S^n$ (pointed!), this procedure gives the Thom space. So we learn that: the Thom space of a pointed spherical fibration is the homotopy colimit taken in the homotopy theory of pointed spaces of the family of spheres indexed over the base.
Example. Given an unpointed spherical fibration $E \to B$, with fibers $S^{n-1}$, we can form the fiberwise suspension by taking the cofiber $S^{n-1} \to CS^{n-1}$, fiberwise. This produces a pointed spherical fibration, and we can then take the Thom space as above. It's not hard to see that collapsing the section at $\infty$ of this suspended fibration gives the same answer as taking the mapping cone of the projection for the original fibration, whence the connection with your first definition.
Example. If $B = BG$ is the classifying space for a group, then the 'homotopy colimit indexed by $BG$' is also called the 'homotopy orbits for the action of $G$', since we may identify a bundle $E \to BG$ with a homotopy coherent action of $G$ on the fiber over a fixed basepoint of $BG$. So, in this case, the Thom space is of the form $S^n_{hG}$ where $G$ acts in some way on $S^n$. This includes the universal example, when $G = \mathrm{hAut}(S^n)$, the space of homotopy automorphisms of $S^n$ (i.e. the degree $\pm 1$ components of $\Omega^nS^n$ when $n>0$).
Thom spectra
For bundles of spectra it is nice to take the 'diagrams indexed by $B$' point of view as a definition. So a bundle is given by a map $B \to \{\text{space of spectra equivalent to }S^0\}=\mathrm{BGL}_1(S^0)$. This, in particular, defines a diagram of spectra and its homotopy colimit is the Thom spectrum. Since $\Sigma^{\infty}_+$ commutes with taking homotopy colimits (it's a left adjoint), we see that taking the suspension spectrum of a Thom space gives the Thom spectrum of the bundle obtained by taking fiberwise suspension spectra.
Now, in the case that $B = BG$, then we may identify functors $BG \to \mathsf{Sp}$ with modules over the $\mathbb{E}_1$-ring $S^0[G] := \Sigma^{\infty}_+G$. (This is a spectral version of the relationship between $G$-modules and $\mathbb{Z}[G]$-modules for an ordinary group). So we have:
spherical fibrations over $BG$ $\iff$ a coherent action of $G$ on the sphere spectrum $S^0$ $\iff$ an $S^0[G]$-module structure on $S^0$.
Under this correspondence, taking homotopy colimits corresponds to the construction on modules $M \mapsto M \otimes_{S^0[G]}S^0$ where we use the augmentation. (This is analogous to the statement that the coinvariants of a $G$-module are computed via a similar tensor product in the classical setting).
Thus, in the case $B = BG$, we may compute Thom spectra as $S^0 \otimes_{S^0[G]} S^0$ where the left $S^0$ has an interesting module structure, and the second $S^0$ is acted on through the augmentation.
Of course, the action of $G$ factors through the largest action of all of $\mathrm{GL}_1(S^0)$, so we can write this as
$S^0 \otimes_{S^0[G]}S^0 = S^0 \otimes_{S^0[\mathrm{GL}_1(S^0)]} S^0[\mathrm{GL}_1(S^0)] \otimes_{S^0[G]} S^0$.
In the special case when $G \to \mathrm{GL}_1(S^0)$ is 'normal', ie arises as the fiber of an $\mathbb{E}_1$-map $\mathrm{GL}_1(S^0) \to H$ (which happens int he infinite loop space context if we take $H = \Omega^{\infty}Cj$ in your notation), then we may simplify the right hand side of the tensor product as
$S^0 \otimes_{S^0[\mathrm{GL}_1(S^0)]} S^0[\mathrm{GL}_1(S^0)] \otimes_{S^0[G]} S^0=S^0 \otimes_{S^0[\mathrm{GL}_1(S^0)]} S^0[H]$.
So that's the relationship between the two definitions you write down.
Best Answer
Ok, this discussion has grown beyond the level of comments so I'll collect the facts here. A bit of terminology: a $(-1)$-connected space is a space with a choice of basepoint and the category of $(-1)$-connected spaces $Top_{>-1}$ is the category of pointed homotopy types with basepoint-preserving maps. The category of $n$-connected spaces $Top_{>n}$ is the full subcategory of $Top_{>-1}$ of spaces $X$ which have all homotopy groups $\pi_k(X) = 0$ for $k \leqslant n$. For any space $X$ we can take its $(n-1)$-truncation $X_{<n}$ which has all homotopy groups $\pi_k = 0$ for $k\geqslant n$. There is a fiber sequence $X_{\geqslant n} \to X \to X_{<n}$ with the right arrow an isomorphism for $\pi_{k<n}$ and the left arrow an isomorphism for $\pi_{k\geqslant n}$.
Now, informally a loop space $X$ is just a space with a homotopy equivalence $X \simeq \Omega Y$ for some other space $Y$, however this definition is not precise enough since there can be many non-equivalent choices of $Y$. We would want some structure on $X$ that would guarantee it is a loop space without explicitly specifying such $Y$. A famous theorem of May tells us that that a space is a loop space if an only if it is grouplike and is equipped with the structure of an algebra over the $A_\infty$ (aka $E_1$) operad, in particular it is $(-1)$-connected. In this case there is a canonical choice of delooping denoted $\mathbb B X$ which is a 0-connected space such that $\Omega \mathbb B X \simeq X$ as homotopy types and as $A_\infty$-spaces. Even more specifically, the functors $\Omega$ and $\mathbb B$ are well-defined on the categories of $(-1)$-connected spaces and $A_\infty$-spaces respectively and establish an adjunction $\mathbb B \dashv \Omega$ between them, which restricts to an equivalence between the full subcategories of grouplike $A_\infty$-spaces and $0$-connected spaces. In particular, $\mathbb B X$ is the unique 0-connected delooping of $X$, and for any other delooping $Y$ its 0-connected cover $Y_{>0}$ is thus equivalent to $\mathbb B X$. Thus we don't need to consider non-connected deloopings when defining loop spaces.
So formally we can define a loop space as a grouplike $A_\infty$-space, and a map of loop spaces as the map of spaces preserving the $A_\infty$-structure (grouplikeness is a property rather than structure so doesn't impose extra conditions). These are exactly the maps induced on the loop spaces from the maps $Y \to Y^\prime$. Note that there can be many different $A_\infty$ structures on the same topoolgical space, thus many nonequivalent deloopings (a classic example is $S^3$ which have different $A_\infty$ structures different from $SU(2)$ group structure arising from mixing various structures at primes). For this reason it doesn't make sense to just say that a loop space is $X$ such that there exists $Y$ with $\Omega Y \simeq X$ --- the specific choice matters.
Generalizing to all $n$, an $n$-fold loop space can be delooped $n$ times and the category of $n$-fold loop spaces is equivalent to the category of $(n-1)$-connected homotopy types. An $\infty$ loop space is thus a space $X$ such that it can be delooped an arbitrary number of times. We can either define it as an algebra over the $E_\infty$ operad (which is a colimit of all $E_n$ with $E_{n+1} = E_n \otimes E_1$) or as a sequence of spaces such that $n$-th space is $(n-1)$-connected and $Y_n = \Omega Y_{n+1}$. Such sequences are exactly connective spectra.
The definition via an arbitrary sequence $Y_i$ requries a different notion of equivalence, since such sequences represent arbitrary spectra. We can say that a map of two spectra $Y_i$ and $Y_i^\prime$ is an equivalence if it is a homotopy equivalence on the 0-th component. Note that the sequence of $(i-1)$-connected covers $(Y_i)_{>i-1}$ is also a spectrum and its canonical map to $Y_i$ is an equivalence of $\infty$-loop spaces. This construction realizes the subcategory of connective spectra as the localization of the category of all spectra. I understand this is Adams's approach in Infinity loop spaces.