Grammatically speaking, as the earlier action described in the sentence ("seeing the movie") was already completed prior to the second action ("Tom inviting me"), and the second action is in the past, only the past perfect tense is acceptable. That is, only "I had already seen the movie" is correct.
However, you may find some native speakers who use "I already saw the movie" in this situation. As I am sure is also true with Japanese, some types of grammatical mistakes "sound wrong", and educated native speakers would be unlikely to say them, while other types of mistakes sound only slightly awkward, and some native speakers might sometimes say them. In this example, the latter version sounds wrong, but not horribly wrong. Without question, though, you should teach your students that only "I already saw the movie" is correct in this situation.
Of the four sentence endings you asked about, (1) and (3) sound the most natural. (4) is grammatically correct, but wordy and a little redundant, so native speakers would be unlikely to use it. (3) is awkward, and grammatically questionable, and should be avoided.