Maybe not a fallacy, but...

To answer you properly, we would need to shed some light on the relation between x and x'. In any case, someone who speaks in this way would quickly bring the opponent to the edge of his nerves.

I think you have hit upon something important here. It is one of the biggest problems I see in attempting to use logic in debates. What several commentators have called this is "moving the goalposts," but that suggests a formal debate or other arena where points scoring is important. In most of these cases, the debate is an attempt to convince or persuade someone a strongly held belief is (in part or in whole) illogical. In such cases, motivated reasoning causes the person upon whom you are attempting to use logic to be 'slippery' with his/her definitions -- as slippery as is needed to avoid admitting the point you are making is correct.

Sometimes this 'slipperiness' occurs in small ways (since all words have ambiguous connotative meanings) and sometimes it occurs in much larger ways (such as completely redefining words). I imagine this is how we got to the situation where if you look up the word "literally" in the dictionary you will find it can mean the  exact opposite of its original and obvious definition (i.e. it can mean "figuratively").

Using your example, imagine trying to debate someone who asserts X is literally true, you show that it is not true (or inconsistent with the facts) and then that person responds they meant it was figuratively true since "literally" can also mean "figuratively"!

I'm not familiar with Sagan's example, but thanks for sharing it. I will look it up. As a Theist, I have also experienced this problem in debating with Atheists. Indeed, I had a debate with an Atheist on this platform that consisted in part of debating the proper meaning of almost every key word that was used. It was quite irritating, so I can see from where you are coming.

[To Jordan Pine]

Update: Sagan's "Dragon in My Garage" analogy can be found here for others who may be interested. In the entry, it describes the tactic in question as applying "increasingly ad hoc reasoning" to justify an argument.

As the Website BigThink writes: "An ad hoc argument isn't really a logical fallacy, but it is a fallacious rhetorical strategy that's common and often hard to spot. It occurs when someone's claim is threatened with counterevidence, so they come up with a rationale to dismiss the counterevidence, hoping to protect their original claim."

Nicely put. Perhaps that is the final, correct answer to your query. What you are describing is not a logical fallacy but a "fallacious rhetorical strategy" that is called "ad hoc reasoning" or an "ad hoc argument."

Prove to Person 1 that his assertion is inconsistent, and you satisfy him for that round of the discussion. Teach him how to assess his claims for consistency, and you satisfy him for the rest of his life and you can stop arguing with him ad infinitum.

Yep, I'd say this or hedging.