My (very) personal understanding of this issue is related to the (not obviuos) concept of Logical Constants.
According to the "traditional" view :
The most venerable approach to demarcating the logical constants identifies them with the language's syncategorematic signs: signs that signify nothing by themselves, but serve to indicate how independently meaningful terms are combined.
In my reading, Enderton uses "logical symbols" with the same meaning of "logical constants", i.e. symbols like $\lnot$ or $=$ that do not change meaning according the context.
Obviously, the meaning of a predicate or constant symbol is specified only through an interpretation.
If we follow this approach, we can say that also the quantifiers are specified through an interpretation; the meaning of $\forall$ is obviously "all", but this "all" changes if we are speaking of natural numbers : in this case $\forall$ means "for all $n \in \mathbb N$", or if we are speaking of Greek philosophers : in this case $\forall$ means "Plato and Aristotle and ..."
But you can see other equally authoritative textbooks (like van Dalen's one) where a similar distinction is not present, and there is only one list of symbols.
You can see also :
The logical symbols will have a fixed interpretation. In particular, “$=$” will always be expected to mean equals.
Regarding variables, we can compare :
proper names [i.e. constants] and variables [ ...] we call proper symbols,
and we regard them as having meaning in isolation, the primitive names as denoting (or at least purporting to denote) something, the variables as having (or at least purporting to have) a non-empty range. But in addition to proper symbols there must also occur symbols which are improper - or in traditional (Scholastic and pre-Scholastic) terminology, syncategorematic - i.e., which have no meaning in isolation but which combine with proper symbols (one or more) to form longer expressions that do have meaning in isolation. [...] Connectives are combinations of improper symbols [...].
we divide all our signs into logical and descriptive (or non-logical). Descriptive signs are those constants which serve to refer to objects, properties, relations, etc., in the world; they include the individual constants, the predicates, and the sentential constants. Logical signs include all the variables and the logical constants. Logical signs do not themselves refer to something in the world (the world of things has nothing like negation, disjunction, etc.); rather, they bind together the descriptive constants of a sentence and thereby contribute indirectly to the sense of a sentence.