The definition of logic is: the science of thinking about or explaining the reason for something using formal methods.
And formal means very correct here?
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That definition of logic is not too bad. However, in that context, "formal" means taking only form into account.
Whether or not logic is a science is a deep, philosophical question. For example, one of the science definitions you linked to is "knowledge about or study of the natural world based on facts learned through experiments and observation". Consider a common property of logic -- that both a statement and its converse cannot be true. Is that "knowledge about ... the natural world"? Do we learn that by observation?
Really, it gets down to semantics. Logic is what it is. If you want to call that part of science, then fine. If not, then not. But it's silly to try to parse definitions that way. Definitions are explanations, not tests you can apply. A sensible definition of "car" could begin "a means of transportation", but a toy car is certainly a car even though it's not a means of transportation.