If deja vu means already seen or experienced, what would you call the feeling that you're experiencing something again, but in fact you haven't; it's just a strong, but false, feeling of familiarity?
That the phrase means "already seen" in French is irrelevant. In English there is no 'false' or 'true' sense of déjà vu. The phrase is used in English to designate the fleeting and imprecise sense that something (which you cannot name) about what you have just seen or heard is familiar, that there was something there which you have heard or seen before. It feels like a memory, but it's not attached to anything specific.
If you were able to pin the memory down sufficiently to be able to declare it false or true, it would not be a sense of déjà vu— it would be a memory. Memories can be false or true, but déjà vu is a state—it has no truth value.
TrevorD points out—and very thoroughly documents—a more literal sense of déjà vu which has become current. I think this arose as an ironic use of the phrase—Does it feel like we've seen this before?—but it may very well be that it is now innocent of that overtone and now means just Here we go again.
It appears that even authoritative dictionaries do not agree on whether proper use of the term deja vu is restricted to events that have not actually happened previously, as shown by the following definitions:
Definitions restricted solely to events that have not actually happened previously:
the feeling or illusion that one has experienced something before although one is actually experiencing it for the first time.
the experience of perceiving a new situation as if it had occurred before.
Collins English Dictionary - accessed from Dictionary.com
Ambiguous definitions referring only to events that have not actually happened previously (but with no explicit restriction):
the feeling that what is happening now has happened before in exactly the same way:
- a strange sense of déjà vu
Longman English Dictionary Online
the feeling that you have previously experienced something which is happening to you now:
- I had a strong sense of déjà vu as I entered the room.
Oxford Advanced Learner's Dictionary
The strange sensation that something one is now experiencing has happened before:
I knew I had never been in the house before, but as I walked up the staircase, I got a weird sense of déjà vu.
The American Heritage® New Dictionary of Cultural Literacy - accessed from Dictionary.com
Definition explicitly not restricted to events that have not actually happened previously:
A feeling of having previously experienced something, especially when that is not the case.
- Have I done this before? Talk about déjà vu.
Definitions referring both to events that have not actually happened previously, and to unwelcome repeat events
a feeling of having already experienced the present situation:
- a feeling of déjà vu
- to list the opponents of his policies is to invite boredom and a sense of déjà vu
Oxford Dictionaries Online (ODO)
the illusion of having previously experienced something actually being encountered for the first time.
disagreeable familiarity or sameness:
- The new television season had a sense of déjà vu about it—the same old plots and characters with new names.
Based on the Random House Dictionary - accessed from Dictionary.com
the illusion of remembering scenes and events when experienced for the first time:
a feeling that one has seen or heard something before:
- I entered the room and immediately felt a sense of déjà vu.
something overly or unpleasantly familiar (chiefly US, informal):
- When the car broke down again, it was déjà vu.
- The rise in housing costs is déjà vu all over again.
Merriam-Webster and Merriam-Webster Learner's Dictionary
In my experience, and based on the above definitions, I would say that:
The primary meaning of the term is the feeling or illusion that one has experienced something before although one is actually experiencing it for the first time.
The term is also used to refer to a repeat of unwelcome past events: something overly or unpleasantly familiar; but I would also say that this meaning is not restricted to the US (as one of the definitions suggests), but is also a not uncommon British usage.
The term is not generally used to refer to actual repeat events, except in the sense of my previous bullet point.
To address the comment from RegDwighт, breaking the question into two parts:
If deja vu means already seen or experienced, ...
- Deja vu does not generally mean already seen or experienced - see paragraph 3. of my answer above - except when used (e.g. sarcastically) as described in my paragraph 2. above.
... what would you call the feeling that you're experiencing something again, but in fact you haven't; it's just a strong, but false, feeling of familiarity?
- That feeling is described as deja vu - see my paragraph 1. above.
Deja-vu is defined as the feeling you are doing something you have done before.
It is important to note that it is the feeling that what you are doing is something you've previously done. Not, necessarily, that the feeling is of a thing you genuinely have done before.
Though if you want to differentiate the two, you could call it "false deja-vu", meaning your feeling is incorrect - you feel as though you have done it before, but that feeling is false.
In French "déjà vu" has two meanings.
- one has been imported into English :
"J'ai une impression étrange de déjà vu" = I have a strange feeling of having already seen something like that, perhaps in a dream".
- the other is strongly pejorative :
"Ses tableaux ? Il n'a rien inventé, rien que du déjà vu !" = About his paintings ? He has invented nothing, everything has been done before.