Any given typing judgement we might want to make about a particular term can be checked statically or not, and it can be checked dynamically or not.

If the language has terms (... that might be evaluated) that don't get checked statically or dynamically, then it is going to be weakly typed (with respect to those typing judgements we're considering).

If we've already checked something statically, we "know" that it will always hold, and checking it dynamically is in some sense redundant (provided we actually believe our checking to be correct and our environment to be sufficiently robust that things don't change out from under us) and I think that's a part of the reason the two feel opposed, but in principle you could check a property both statically and dynamically for a particular term.

Step back a little to the language level and they aren't at all opposed; you might want to check some properties statically and some dynamically (eg. static checking of class/interface vs dynamic nullability in Java) or check a given property statically for some terms and dynamically for others (eg. gradual typing).