To add to Keenan's answer and hopefully address the thinking in the OP a little more explicitly:
$F[x]/(x^2)$ is, as you say, the ring you get when you adjoin to a field an element $x$ satisfying $x^2=0$. And, as you say, $0$ already satisfies this relation in $F$. I interpret the second paragraph of the question to be asking, "Doesn't this mean $F[x]/(x^2)$ is just $F$ adjoin $0$?" Another way to ask this could have been: "Is $F[x]/(x^2)$ the same thing as $F$?"
The answer is no, because $x\neq 0$ in $F[x]/(x^2)$. Thus this ring has another element besides zero whose square is zero. Here are two ways to look at why this is true:
(1) By the definition of $F[x]/(x^2)$, it is the ring of additive cosets of the ideal $(x^2)$. The zero coset is the ideal $(x^2)$ itself. Now $x$ is not in this ideal; that means the coset $x+(x^2)$ is different from zero in this quotient ring.
This "nuts and bolts" argument has the advantage that it connects directly to the definition, but the disadvantage that it doesn't address the way of thinking about $F[x]/(x^2)$ that led you to wonder if $x$ was zero. Hopefully that is addressed by -
(2) As you said, we can think of $F[x]/(x^2)$ as $F$ adjoin an $x$ such that $x^2=0$. But this ring is not just some ring containing $F$ and an element $x$ whose square is zero. It's the (unique) "biggest"/"freest" ring that has this. The only relations that it has between $x$ and other elements of the ring are those forced by $x^2=0$ (such as $x^5=0$, $x^2+1=1$, etc.). In a ring (unlike a field), it is possible for an element to square to zero without equaling zero, so for "maximum bigness/freedom", the element $x$ that is adjoined to $F$ in $F[x]/(x^2)$ is different from zero. $x^2=0$ does not force $x=0$ so $F[x]/(x^2)$ does not have this latter relation.
This second argument is an informal statement of the fact that the quotient $F[x]/(x^2)$ is characterized by a "universal property": for any ring $R$ containing $F$ and also an element (say $\alpha$) whose square is zero, $F[x]/(x^2)$ has a unique homomorphism to that ring that is the identity on $F$ and sends $x$ to $\alpha$. Since it has to be able to do this for any ring $R$ with these properties, it can't have any "extra" relations.