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In sentences #1, 'really' acts as an intensifier. Using "teacher talk", I would say that 'really' softens the sentence, but I can't think of a technical term for it. What would that be? Is it simply "qualifier"? Is there another word for the opposite of "intensifier"?
An approximately opposite term to intensifier is downtoner (adverbs like barely, hardly, fairly, relatively, etc. are sometimes categorized as such, e.g. by Quirk et al. A Comprehensive Grammar of the English Language, Longman 1985, pp. 445ff), but I do not think you need to look for the opposite of "intensifier" if what you are trying to do is label the semantic function of really in your example 2.
The difference between I like apples and I really like apples is, indeed, one of 'strength' or 'emphasis', and saying that in that context really is an 'intensifier' or an 'emphasizer' (as Quirk et al., op. cit., p. 447, prefer to call it) is appropriate, I suppose, to the extent such functional-semantic labels can be defined at all.
However, what matters to the present issue is that speakers would normally never use that adverb unless they had a reason to 'strengthen' their statement, and the obvious one is that a previous statement of theirs to the same effect, but without the adverb (i.e., I like apples), has not, in their view, been strong enough.(Or, if the sentence is uttered out of the blue, that some contextual factor has induced the addressee to believe that the speaker does not like apples, or not enough). Hence, I really like apples can be expected to occur mostly as a 'corrective' statement (the correction consisting in strengthening a previous statement not considered effective enough or in neutralizing the contextual factors that may have led the addressee to the wrong belief that the contrary is the case).
Correspondingly, whereas I don´t like apples is a remark you might also make out of the blue, I don´t really like apples is not likely to occur except to 'correct' a previous statement to the contrary (or, again, neutralize some contextual factor inclining the addressee to believe that the speaker, in this case, likes apples or likes them especially). Hence, the function of really is basically the same as above: 'intensification' and 'correction', although, in this case, the lion's share of the correction falls to negation itself, because the crucial difference between the positive and the negative example is that, in I don´t really like apples, the correction is not a matter of strengthening an earlier weak statement, nor one of 'softening' a strong one to make it more polite; it is a matter of cancelling an earlier statement without too much embarrassment. What really does in that context is help negation deny, without making the speaker violate the Gricean principle of Quality in a flagrant way, a previous false statement of his to the contrary (i.e., an unmarked I like apples). Your intuition that it has a 'softening' effect is probably right, but can be explained by the fact that negating an 'intensified' statement is not as violent a self-correction as negating a non-intensified one, since it is compatible with an interpretation in which only the intensification is removed, while the basic propositional content is preserved (i.e., imagine that what the speaker has in mind is I said earlier that I liked apples - or something has led the hearer to believe that I do - but I did not mean that I particularly like them, although I do not plainly dislike them either, so it is not that I don´t like them, either, so I did not properly lie earlier. That would avoid the embarrassment of flatly admitting that his earlier statement was not true).
In sum, in my view, you need not look for an opposite term to 'intensifier' to account for the function of really in 2. By itself, really probably has the same function in both examples, that of an 'intensifier'. It is only when it is under the scope of negation that it may acquire a 'softening' role, but only in the sense, and for the reasons, already explained.
Mitigator: Mitigate v. make less severe, serious, or painful: he wanted to mitigate misery in the world. • lessen the gravity of (an offense or mistake): (as adj. mitigating) : he would have faced a prison sentence but for mitigating circumstances.
