selective deletions from list

SequenceReplace[lis, {OrderlessPatternSequence[{a_, "x", c_}, {a_, b_, c_}]} :> {a, b, c}]

{{"b", "x", "d"}, {"a", "z", "b"}, {"a", "x", "c"}, {"b", "z", "d"}}

One approach is to look at the differences between adjacent elements, find those which equal {0,0,x_}, and remove them from the list.

lis[[Complement[Range[Length[lis]],Flatten@Position[Differences[lis], {0, 0, x_}]]]]

{{"b", "x", "d"}, {"a", "z", "b"}, {"a", "x", "c"}, {"b", "z", "d"}}