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Why is a symmetric relation defined by $forallxforally(xRy implies yRx)$ and not $forallxforally(xRy iff yRx)$?
(I have only found a couple of sources that defines it with a biconditional)

For example, according to Wolfram:

A relation $R$ on a set $S$ is symmetric provided that for every $x$ and $y$ in $S$ we have $xRy iff yRx$.

But the majority of books defines it the other way.
And I think I agree with the second definition.

Because if we use the first definition with "$implies$", we know the truth table of the implication in particular $P implies Q$ is true when $P$ is false and $Q$ is true. That means in the context of symmetric relation that $(x,y) notin R implies (y,x) in R$ is true.

And the example $A = 1,2,3,4$ with relation $R = (2,1),(3,1),(4,1)$ satisfies the definition because $(x,y) notin R implies (y,x) in R$ is true.

And for me it's weird that this case is considered symmetric.
Or maybe I have a profound confusion with the concept.
I would like that you guys help me clarify. *Sorry for my grammar I'm not a native english speaker.

For all $x$ and all $y$ make the if and only if unnecessary (albeit perfectly acceptable).

1) $(x,y) in R implies (y,x) in R$ for ALL $x,y in A$

And the statement 2) $(x,y) in R iff (y,x) in R$ are equivalent statements.

If 1) is true and $(x,y) not in R$ then although $(x,y)in Rimplies (y,x)in R$ or $F implies (y,x)in R$ is true, it does not tell us any thing about whether or not $(y,x) in R$. However $(y,x) in R implies (x,y) in Y$ tells us that $(y,x) not in R$. Because $(y,x) in R implies (x,y) in R$ means $(y,x) in R implies F$. An the only thing that implies a false statement is a false statement. So we must have $(y,x) not in R$.

So in your example you have $(1,2)in Rimplies (2,1)in R$ is true but you don't have $(2,1) in R implies (1,2) in R$ as true.

So it isn't symmetric.

=====

Another way to look at it:

If $A = 1,2,3$

Then we will have 9 statments.

By 1) the nine statements are:

$(1,1)in Rimplies (1,1) in R$

$(1,2) in R implies (2,1) in R$

$(1,3) in R implies (3,1) in R$

$(2,1) in R implies (1,2) in R$

... etc... all nine are needed.

With 2) we also have nine statements:

$(1,1)in Riff (1,1) in R$

$(1,2) in R iff (2,1) in R$

$(1,3) in R iff (3,1) in R$

$(2,1) in R iff (1,2) in R$

...etc....

$(1,2) in R iff (2,1) in R$ and $(2,1) in R iff (1,2)in R$ is redundant.

So aesthetically, using definition 2) is .... inefficient.

If $A$ is a set and $R$ is a binary relation defined on $A$ (that is, $R$ is a subset of $Atimes A$), then the usual definition of symmetry (as far as $R$ is concerned) is$$(forall xin A)(forall yin A):xmathrel Ryimplies ymathrel Rx.tag1$$And this is equivalent to$$(forall xin A)(forall yin A):xmathrel Ryiff ymathrel Rx.tag2$$So, why do we choose $(1)$ instead of $(2)$ in general? Because, in general (although not in this case) it is easier to verify the assertion $Aimplies B$ than $Aiff B$. And (again, in general), when we choose between two distinct but equivalent definitions, we usually choose the one which is easier to verify that it holds.

If $xRy implies yRx$ for all $x$ and all $y$, then we can choose $x := tildey$ and $y := tildex$ and get $tildeyRtildex implies tildexRtildey$ or, equivalently, $yRx implies xRy$, which is $impliedby$.

In conclusion, since the implication should hold for all $x,y$, the equivalence already holds.

Let it be that $R$ is a symmetric relation.

This according to the first mentioned definition:$$forall xforall y[xRyimplies yRx]tag1$$

Now let it be that $aRb$.

Then we are allowed to conclude that $bRa$.

On the other hand if $bRa$ then also we are conclude that $aRb$.

So apparantly we have:$$aRbiff bRa$$

Proved is now that for a symmetric relation $R$ (based on definition $(1)$) we have:$$forall xforall y[xRyiff yRx]tag2$$

So $(2)$ is a necessary condition for $(1)$.

Next to that it is obvious that $(2)$ is also a sufficient condition for $(1)$ so actually the statements are equivalent.

Both can be used as definition then, but in cases like that it is good custom to go for the one with less implications. This for instance decreases the chance on redundant work if we try to prove that a relation is indeed symmetric.