Short Answer
The answer to your question is at the start of the article you cited:
Each logical system in this class shares [these 5] characteristic properties... While not entailed by the preceding conditions [emphasis mine], contemporary discussions of classical logic normally only include propositional and first-order logics. Classical logic generally entails the majority of the Western tradition of philosophy and logic since the pre-Socratics, but particularly starting with Aristotle.
"While not entailed by the preceding conditions" means that 'there is more than these 5 conditions to describe classical logical study'. Five sentences would hardly be enough to describe classical logic.
Long Answer
What is classical logic? According to the Encyclopedia of Philosophy and it's entry 'Logic, Non-Classical':
[C]lassical logic [is] the theory of validity concerning truth functions and first-order quantifiers likely to be found in introductory textbooks of formal logic at the end of the twentieth century.
One way to define what classic logic is is to define it as what non-classic logic is not. (See PhilSE: In how many and which ways can a logic be non-classical? Are there systems for organizing them? for a list of systems and their characteristics.) This is why the principles are featured prominently in this article because what non-classic logic is about is deviation from some of these principles. For instance, sequent calculus and its extensions such as display calculi, nested sequent systems, and labeled sequent systems are metalogical and go beyond classic-logic as metalogical theory. But it's much more informative to describe some of the features of classic logic and some of their historical origin.
Is there more than the principles? Well as the WP article concedes, yes. In fact, of the three generally acknowledged Laws of Thought, one is not mentioned in the list of principles above: the Law of Identity. So, immediately, the moment discussion starts about classic logic, the Law of Identity is an important topic; the famous philosopher of language, John Searle, for instance, challenged the Law of Identity as a law arguing it is only a convention in his paper Proper Names. (See PhilSE: On Searle's _Proper Names_ (1958) for an explanation how he invokes the conventions of cryptography to challenge it.)
From Volume 5 of the Encyclopedia of Philosophy, the entry "Logic, Traditional" begins with this:
In logic, as in other fields, whenever there have been spectacular changes and advances, the logic that was current in the preceding period has been described as "old" or "traditional"...In every case, the logic termed "old" or "traditional" has been essentially Aristotelian, but with a certain concentration on the central portion of the Aristotelian corpus, the theory of categorical syllogism... especially of the sixteenth to the nineteenth century.
In fact, WP's entry on the History of Logic has a subsection on traditional logic:
Other works in the textbook tradition include Isaac Watts's Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection[87] influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany.[88]
So, besides missing the Law of Identity, here are some additional topics: categorical syllogism, the logic of propositions, equipollence, and Euler's diagrams, as well as the history of classical logic more generally to understand the context of their development and relations. In fact, two of the most important works are from three heavyweights of the analytical tradition. According to that WP article:
Classical logic reached fruition in Bertrand Russell and A. N. Whitehead's Principia Mathematica, and Ludwig Wittgenstein's Tractatus Logico Philosophicus.
A more technical overview can be found at SEP: Classical Logic.