To some extent this is a question about word usage in mathematics. One natural reading of the English term "weakly connected" would be "the connectedness of the graph is weak", weak meaning that it is deficient in some sense like a weak signal.
However, the use of "weakly" here is more of a conjugation of the term "weak connectedness", i.e. connected in a weak sense of the word "connected", with weak meaning that it is not a very strenuous requirement. A graph is weakly connected if it satisfies this weak connectedness requirement, not because it is less strongly connected than other graphs.
This is a fairly common idiom in math. In functional analysis one might refer to a sequence that is "weakly convergent" (again meaning convergent in a more generous sense that has a precise definition), and then later show that the same sequence is in fact strongly convergent.
There are other cases where the terminology is indeed exclusive, unlike the present case: when an infinite series is "conditionally convergent", the definition specifically excludes the case that it is absolutely convergent (which is a stronger notion of convergence).
