Why is one relation transitive but the other is not?

To be transitive, $xRz$ needs to hold whenever you have $x$, $y$, and $z$ such that $xRy$ and $yRz$. You're only testing the case $x=y=1$ and $z=3$, but there might be other cases. For instance, in your first example, you can let $x=1$, $y=3$, and $z=2$, and you find that $1R3$ and $3R2$, but $1R2$ is not true. So the relation is not transitive.

(By the way, you should not call that first relation an "equivalence relation", since an equivalence relation is required to be transitive, in addition to being reflexive and symmetric.)

$xRy$ and $yRz$ should imply $xRz$ for all choises of $x,y,z$.

In your case you have just noticed that $1R3$ hold in both examples. This is however not sufficient for transifitivty, you have to check all possible cases of $x,y$ and $z$ in order to show that a relation is transitive.

In the case of $R =\{(1,1),(2,3),(1,3)\}$, the only tuples where both $xRy$ hold and $yRz$ is when $x=1$, $y=1$ and $z=3$ or when $x=1$, $y=1$ and $z=1$. If we choose $x,y$ or $z$ in any other way then either $xRy$ will not hold or $yRz$. Now it is straight forward to check that if $x=1$, $y=1$ and $z=3$ then $xRy,xRz$ and $xRz$ and if $x=1$, $y=1$ and $z=1$ then $xRy,xRz$ and $xRz$.

Now for the other relation $R = \{(1,1),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\} $, choosing $x=1,y=1,z=3$ does not pose a problem as $1R1,1R3$ and $1R3$ hold. However If we choose $x=2,y=3,z=1$ then we see that $2R3$ and $3R1$ both hold, however $2R1$ does not hold (since $(2,1)\notin R$) which was supposed to hold in order for the relation to be transitive. Thus this gives one example of where the relation "$xRy$ and $yRz$ imply $xRz$" does not hold, and thus the relation is not transitive.

A relation $R$ of the set $S$ is transitive if: $$\forall a{\in} S~\forall b{\in}S~\forall c{\in}S: \Big(\big((a,b){\in}R\wedge(b,c){\in}R\big)\to (a,c){\in}R\Big)$$

That definition is equivalent to:

$$\neg \exists a{\in}S~\exists b{\in}S~\exists c{\in}S: \Big(\big((a,b){\in}R\wedge(b,c){\in}R\big)\wedge (a,c){\notin}R\Big)$$

Thus what you are looking for are counter examples.   Look for any $(a,b), (b, c)$ in the relation without a matching $(a,c)$ in the relation.   Just one shows the relation is not transitive, but you have to be sure there are none to claim transitivity.

My first example is a "equivalence relation" $S=\{1,2,3\}$ and $R = \{(1,1),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\}$ My Book solutions say that this relations is Reflexive and Symmetric

This is not transitive because while $(1,3)$ and $(3,2)$ are in the relation, $(1,2)$ is not.   One counter example is all we need.

PS: Because this is not transitive, it is not an equivalence relation.   Equivalence relations are those which are reflexive, symmetric, and transitive.

My Second example is "partial order" $S=\{1,2,3\}$ and $R =\{(1,1),(2,3),(1,3)\}$ My Book solutions says is Antisymmetric and Transitive

This is transitive because there is only one pair of $(a,b),(b,c)$ elements from which a counter example could be formed, $(\color{red}1,\color{blue}1),(\color{blue}1,\color{red}3)$, but $(\color{red}1,\color{red}3)$ is indeed in the relation; so there is no counter example.