One book defines the maximal element as
“Let $(P,<)$ be a poset. An element $c\in P$ is called maximal if for all $a\in P$ with $c<a$ we have $a=c$.“
How can one write $a=c$, for a is all the elements of $P$ isn’t it? How can $c$ be equal to all of those different elements?
On
A maximum element $m$ is such that $\forall x \in P: x \le m$.
A maximal element $m$ is such that $\forall x \in P: (x \ge m) \to (x=m)$.
So if in element is higher than $m$ it can only be $m$ itself (I prefer to use $\le$ for posets as $<$ for me denotes strict orders so that $a < a$ is never possible). So you cannot go "beyond" $m$. There might be many maximal elements without there being a maximum (e.g in the poset of all linearly independent subsets of a vector space $V$, ordered by inclusion, the maximal elements are exactly the different bases for $V$).
When the definition say "all $a\in P$ with $c<a$, we have $a=c$", the $a=c$ part is only relevant if the antecedent holds.
That is, if $a$ does not satisfy $a<c$, we are not claiming that it is equal to $c$.
Consider for example the statement "For every prime number $p$ with $p>2$, we have that $p$ is odd". Does it mean that all primes are odd? No.