Your argument up to ''$\ker f=S_3\text{ or }\mathbb{Z}_6$'' is correct.
But after this, it is possible but lengthy to continue the arguments; for example if kernel is isomorphic to $S_3$ then you have taken it equal to $\{(1), (123),..\}$; this is correct but needs a justification.
Better is the following: $|\ker f|=6$, so $\ker f$ contains an element of order $3$. Since elements of order $3$ in $S_4$ are precisely $3$-cycles (easy to prove) and any two $3$-cycles are conjugate, hence all the $3$-cycles of $S_4$ should be in the kernel (since kernel is normal).
But now we get a contradiction. How many $3$-cycles are there in $S_4$? What is size of kernel?
The keep this question from being unanswered...
Let $G$ be a finite group generated by $x$ and $y$, with $x$ and $y$ of order $2$. We want to show that $G\cong D_{2n}$ (the dihedral group of order $2n$), where $n$ is the order of $xy$.
To answer your questions first: in the dihedral group $D_{2n}=\langle r,s\mid r^n = s^2 = 1, sr=r^{-1}s\rangle$, every element not in $\langle r\rangle$ is of order $2$. To verify this, note that every element can be written uniquely as $r^is^j$, with $0\leq i\lt n$, $0\leq j\lt 2$. The elements not in $\langle r\rangle$ are precisely the ones with $j=1$. Such an element satisfies:
$$\begin{align*}
(r^is)^2 &= r^i(sr^i)s\\
&= r^i(r^{-i}s)s &\text{(since }sr=r^{-1}s\text{)}\\
&= r^0s^2\\
&= 1.
\end{align*}$$
Thus, all such elements are elements of order $2$.
When $n$ is odd, these are the only elements of order $2$; when $n$ is even, all of these are elements of order $2$, and so is $r^{n/2}$. So in a dihedral group, you always have at least half the elements of order $2$.
If you think of the dihedral group as the symmetries/rigid motions of a regular $n$-gon sitting on the plane inscribed in the unit circle, you have several axes through which you can reflect the polygon, not just the $x$-axis. The bisection through each vertex gives you a line through which you can reflect the polygon, getting an element of order $2$.
Now, the proof of the desired statement. We note that $xy$ and $y$ satisfy the relations in the presentation of $D_{2n}$: indeed, by definition of $n$ we know that $(xy)^n = 1$; and $y^2=1$ by assumption. Finally, we have that
$$\begin{align*}
y(xy) &= (yx)y\\
&= (y^{-1}x^{-1})y &\text{(since }x^2=y^2=1\text{)}\\
&= (xy)^{-1}y.
\end{align*}$$
By von Dyck's Theorem there is a homomorphism $f\colon D_{2n}\to G$ mapping $r$ to $xy$ and $s$ to $y$. Under this homomorphism, $x$ is the image of $rs$.
(There are other possible homomorphisms, since the map sending $r\mapsto r^i$, $s\mapsto s$, with $\gcd(i,n)=1$, is an automorphism of $D_{2n}$, so pre-composing it with $f$ gives you a slightly different map).
The link you give in your question contains several different answers showing that this $f$ is indeed an isomorphism.
Best Answer
It all seems correct to me. Well done.
As is customary for a proof-verification question, an answer that is more than just a yes/no response is more polite & helpful.
So just let me mention that the Correspondence Theorem is sometimes referred to as the "Fourth Isomorphism Theorem" for reasons such as a series of questions such as yours.