Did the Buddha ever define what he meant by “self”?

Anattalakkhaṇasutta is a short sutta for "self" explanation. 

Brahmajālasutta is a long sutta. 

All abhidhamma, especially paṭṭhāna, are "non self" explanation. This is the longest explanation. 

If you can't understand "self" by all of these sutta, it means your pali skill is lacking.

Self is paññatti. It is not grasping. Also, we choose to be mahāvihāra theravāda buddhist because we want to get all security.

Self is paññatti. It is not grasping. Grasping arise to attach/to cling khandha and paññatti, that are grasping's object. But paññatti never arise, so self never begin or end. Similarly, when you dream. The dreaming-mind is real and arising. But it's object never arise and never vanish, just an imagination of mind. So, self is not real.
Furthermore, for aggregates and grasping, in next life, is already taught that still arise until an ignorant and a craving still is not destroyed by arahanta's enlightenment . Because, there is a future-clinging-aggregate, and that clinging will try (kamma-bhava, kamma-becoming) to have (before death=get new thing, after death=rebirth) that future-clinging-aggregate. So in Khandhasutta taught:

"Whatever form — past, future (form-clinging-aggregate), or present; internal or external; blatant or subtle; common or sublime; far or near — is clingable, offers sustenance, and is accompanied with mental fermentation: That is called the form clinging-aggregate.

And Paṭiccasamuppāda taught:

From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth.

Furthermore, we choose to be mahāvihāra theravāda buddhist (because we using mahāvihāra's tipitaka) because we want to have anytime and anywhere security by make every possible protection to protect every of your future self from every possible insecure. If you still lazy to protect your self from next life/rebirth (by making sure you [five aggregates] must never arise anymore), you are not mahāvihāra theravāda buddhist practitioner. Also, you never understand throughout the theravāda-tipitaka's concept.

WHY a belief in rebirth does not lead to the cessation of suffering?

nihilism ("non existence") and/or existentialism ("existence") must be avoided.

In the middle way, we accept all possible situation such as kamma causes rebirths, next lives, because we believe in causes and effects. But we do not believe in rebirth of self, because it is not the truth. It is just a rebirth of new effects, that may have the same look as in the past, but it is no self at all.

Q: But you're not saying anything different than I am Bonn. When we shed this shell and some other space is entombed in a meat-bag it's not the same, but it is the same. The space awareness was all one - except now it is another space covered by another meat-bag. You're saying the same thing I am yet you guys call my "continuity" to be existence. If MY awareness, that in my shell right now, were to be swapped with that of another shell I would still feel like the same observer but with different memories and perhaps varied senses.

A: I said "It is just a rebirth of new effects", but you said "the same". New is not the same. Your self view make you see the same. But new is new, new is not the old. A cause can not be itself's effect. Self causes not rebirth, self causes not self, kamma causes not self. Just new causes cause, new effects, rebirth. No self at all. Self is just an imagination of mind that is not smart enough to analysis the very fast ( the all time fastest stuff in the universe) of aggregates rebirth. So non practitioner always see "old aggregates" is "the same of new aggregates".

Q: I think you forget the cause and effect chain which results in karma. The end of the chain and its' beginning are the same.

A: No it is not the same kamma. In a second, we made over billion, uncountable, kamma. And in one person depend on uncountable kamma, and change every time. So it is not the same. No self at all. Don't try to keep self view. 

For the example, one's eyes born by wholesome kamma. Then these eyes are blinded by unwholsome kamma, after birth. So is wholesome kamma self? is unwholesome kamma is self? are regular eyes self? are blind eyes self? Or it is just new causes (that maybe arose in the past, too, and it still possible to be a cause of new effects) and new effects arising and vanishing?

This is the reason why paṭiccasamuppāda is very important in buddhism. 

If there is no Self , what Self is there to be affected by the karma which non-Self now performs?

Self in Tipitaka is a general word, It means a wrong view just in no causes/wrong causes context.

Attā (self) can found in both wrong view ('eso me attā'ti), and right view (attāti attano nātho) context.
Context is the definition of the self-wrong view and self-right view.

So by the wrong view, self, manpower self, affects the other self, or is affected by the other self.

But by the right view, self, no manpower and various causes & effects, affect the other self, no manpower, and various causes & effects, or is affected by the other self, no manpower and various causes & effects.

Kamma is just one of many causes of effect.

Effect need more than one causes to arises.

So, perfect manpower self in only one person is not noble truth.