Unfortunately I don't see an "obvious" structure map on the wedge that you describe. It looks more like the smash product for symmetric spectra and doesn't work without the symmetric group action. You'd need to describe maps
$$
S^1 \wedge X_p \wedge Y_q \to \bigvee_{r+s=p+q+1} X_r \wedge Y_s
$$
and it's not clear whether one should use the structure map of $X$ or $Y$ (possibly first applying the twist), or perhaps you had an application of the pinch-multiplication in mind?
So far as the two definitions of smashing with $S^1$, you need to be careful; I'm not sure that both of these definitions are functorial.
In case you hadn't already investigated: for questions about constructing a smash product on these types of spectra, Adams' "Stable homotopy and generalized homology" describes in detail a construction of "handicrafted" smash products by starting with $X_0 \wedge Y_0$ and making a sequence of choices about whether to apply the structure map to the left- or right-hand factor. The proof that it is independent of these choices and gives rise to a symmetric monoidal structure (all after passing to the homotopy category) has been superseded technologically and I wouldn't recommend reading it in detail.
Here is a general fact (due to a bunch of people in the late 60s, early 70s):
A connected space is (weakly equivalent to) an infinite loop space if
and only if it admits an action of an $E_\infty$-operad.
Let me try to unpack this statement. An $E_\infty$-operad $\mathcal{O}$, first of all, is an operad all of whose spaces $\mathcal{O}(n)$ are contractible.
(Sometimes also want to require that the action of the symmetric group on $\mathcal{O}(n)$ be free; this is important if you want the theory of $E_\infty$-algebras to be homotopy invariant. For instance, if you take your $E_\infty$-operad to consist of a point in each dimension, then sure, an algebra over that operad -- that is, a topological abelian group -- is an infinite loop space -- but those algebras tend not to be interesting, and certainly don't model all infinite loop spaces. Note that topological abelian groups are always weakly equivalent to infinite products of Eilenberg-MacLane spaces.)
Now being an algebra over an $E_\infty$-operad is a way of saying that your space is as close to a commutative monoid as possible. That is, there's a multiplication law (pick any point in $\mathcal{O}(2)$ to get a map $m: X^2 \to X$), it's homotopy associative (that's because the two $m(m(\cdot, \cdot), \cdot), m(\cdot, m(\cdot, \cdot))$ are both 3-ary operations coming from $\mathcal{O}(3)$, which is contractible). Moreover, there are higher coherence homotopies (infinitely many) which are conveniently packaged in the operad: that's one of the things operads do efficiently!
One example of a coherence condition is the following. So we know that $m(x, m(y, (m(z, w)))$ and $m(m(m(x, y), z), w)$ are both canonically homotopic (as maps from $X^4 \to X$). But there are two different ways we could make the homotopy go. We want a coherence homotopy between those two homotopies. This is the analog of the MacLane coherence axioms on a monoidal category; you want the various iterated identifications one can make between iterated multiplication laws to be all homotopic.
I'll also mention a weaker (and easier) result:
A connected space is (weakly equivalent to) a loop space if and only if it admits an action of an $A_\infty$-operad.
An algebra over $A_\infty$-operad is something which is supposed to be as close to associative as possible. The standard example is the little intervals operad. If you take $\mathcal{O}(n)$ to consist of the space of imbeddings of $n$ intervals in the interval, then that acts on $\Omega X$ for any $X$. (How? Use these embeddings to compose a bunch of loops.) The point is that being an $A_\infty$-space, rather than simply a homotopy associative H space, is the data you need to construct a classifying space. (The whole story essentially begins with the theory of classifying spaces; it's what shows you that any topological group $G$ is the loop space on $BG$.) You can do this either by first strictifying your $A_\infty$-space into an actual topological monoid (yes, you can do this, essentially because the associatve operad is still "reasonable" insofar as it is acted on freely by the symmetric group; Berger and Moerdijk's paper on homotopy theory for operads), and then take the usual classifying space. Or you can do it directly, e.g. as Segal does it in the paper I mention below.
The result for $E_\infty$-operads and infinite loop spaces is supposed to be more or less the following: take iterated classifying spaces. I don't think that's how May does it (though I don't understand May's construction that well at the moment), but apparently Boardman and Vogt prove it that way.
A very intuitive and fun paper on this sort of thing is Segal's "Categories and cohomology theories." Segal introduces his own form of delooping machinery, which in fact implies the result about $A_\infty$-spaces that I described above.
Finally, let me say something about exactly how
$E_\infty$ spaces are like abelian groups. There is a general theory of algebras and commutative algebras in a monoidal $\infty$-category, due to Lurie (developed in DAG II and III, now in "Higher Algebra"). The definition works out so that "commutative algebra" really means "homotopy commutative algebra up to infinitely many higher homotopies" (as it always does in $\infty$-land). So an associative algebra object in spaces is an $A_\infty$-space, and a commutative algebra object is an $E_\infty$-space. I think one of the motivations of higher category theory is to say efficiently what "up to coherent homotopy" means. For instance, you can think of an $\infty$-category as a "topological category where multiplication is associative up to coherent homotopy."
Best Answer
The homotopy category of spaces is relevant to unstable homotopy theory. All of the other categories are equivalent. If all you want is the stable homotopy category, then Adams' construction is still the most efficient. Note that Adams' construction is just Boardman's, fleshed out. But most people would like a more manageable point-set and/or $\infty$-categorical model for spectra, these days.
EKMM had the first excellent point-set construction of spectra, via so-called $S$-modules. However, it seems to me to be a more or less established consensus, even for the authors of that book, that model categories of diagram spectra are a more manageable "in", if you only want one construction. Diagram spectra describe spectra as continuous functors from some topological category into spaces, where the domain can range in complexity from the discrete category of natural numbers (only identity maps) to the entire topologically enriched category of finite CW-complexes. For smaller domain categories, one has to define spectra as functors equipped with an action of a certain ring-functor $\mathbb{S}$; when $\mathbb{S}$ is a commutative ring-functor one gets the Day convolution symmetric monoidal structure on its modules (very roughly analogous to the tensor product of modules over an ordinary commutative ring) and that gives the nice point-set smash products of spectra to which you allude. As far as where to learn about these things, it's a bit tricky. Adams is still absolutely worth reading. Schwede has an incomplete book project on symmetric spectra that would get you a lot of the basics. Mandell-May-Schwede-Shipley have a nice survey on model categories of diagram spectra to get an overview of what the options are.
While the point-set categories of spectra are very important constructions, one explanation for a relative dearth of thorough expositions on them is that the point-set level details are more important for proving that certain constructions exist than for the majority of actual mathematical work being done in stable homotopy theory. For that, one is often working in the $\infty$-category of spectra, which is really canonically defined: for instance, it's the initial stable $\infty$-category admitting a homotopy colimit-preserving functor from the $\infty$-category of spaces. Every construction of a model category of spectra is a different way of getting at this same $\infty$-category. As for references here, well, stable $\infty$-categories in general are the subject of Lurie's opus Higher Algebra. However, again, many stable homotopy theorists I know rarely think at this level either. A significant proportion of stable homotopy theory is about trying to compute homotopy and cohomology groups, which eventually often reduces to understanding certain spectral sequences in a very concrete and algebraic way. You could look, for instance, at Ravenel's books to get a bit of an idea of this computational perspective.