Weak NP-completeness
As is often the case, the Wikipedia article is misleading.

A problem involving integers parameters is:


Strongly NP-complete if the problem is NP-complete when the parameters are encoded in unary (i.e., $n$ is encoded as $0^n$).
Weakly NP-complete if the problem is NP-complete when the parameters are encoded in binary (which is the usual encoding).


In other words, there is only one notion of NP-completeness, and the confusing terms strongly and weakly NP-complete simply refer to different encodings of problems.

For a problem like SAT, these terms are simply inapplicable, since SAT has no integer parameters. The terms do apply to the related problem of weighted MAX-SAT, in which the input is a set of weighted clauses and a target, and the problem is to determine whether there is an assignment which satisfies clauses whose total weight exceeds the target.