If we can follow sound reasoning such that some statement can be either true or false, based on the exact steps we take, that is exactly the same problem: a contradiction somewhere.
"This sentence is true" doesn't contradict itself, but two different ways of evaluating it contradict each other.
Researcher A assumes the sentence is true and comes to the conclusion that it is.
Researcher B assumes that it is false, substitutes in the value and also concludes that it is correct.
Neither has found a contradiction in isolation, but A and B's findings contradict each other.
In terms of electronics-based intuition, "this sentence is false" is like an unstable circuit: e.g. an inverter feeding its own output to its output. This might result in oscillation. "This sentence is true" is like bistable feedback: it gives us a flip-flop which "latches" whichever value we inject.
Your example is also interesting, but doesn't really break things in the same way.
I don't necessarily agree with this, I think there are plenty of important things to consider with both, but that's why one became famous without the other.