I'm still confused about the two abstract definitions. Can you give some examples to explain?
A classic example of cyclic rule application is stress assignment in Palestinian (and other) dialects of Arabic (Brame (1974) "The Cycle in Phonology: Stress in Palestinian, Maltese, and Spanish"). Stress is assigned to the rightmost heavy syllable, with the provision that a single consonant at the end of a word does not define a heavy syllable, thus kátab 'he wrote', katábit "she wrote" katábna "we wrote". An unstressed high vowel in an open syllable is deleted: this interacts with stress assignment, showing that stress assignment precedes Syncopy: fíhim 'he wrote', fíhmat 'she wrote', fhímna 'we wrote'. A puzzle is that the object clitic -na "us" causes stress to be assigned to the preceding vowel (katábna 'she wrote us', katábna 'he wrote us'), but when the first vowel is /i/, it is not deleted when the subject is 3rd person – fihímna 'he understood us'.
The cyclic explanation is that the form begins as just /fihim/ 'he wrote'. The phonological rules apply to this string and give fíhim. Then the object clitic is added, and the phonological rules re-apply to the combination /fíhim-na/. Stress is re-assigned to the (new) penult, but a vestige of that stress remains on the initial syllable, therefore Syncope cannot apply (because the initial vowel is stressed). This situation where rules apply in the order A-B-A is allowed only in case the rule applies cyclically. The crucial thing about this example where fihim-na has two morphosyntactic analyses ('we understood' vs. 'he understood us') with different outputs is that 'he understood us' is a semantic and structural compositional function of "he understood" and "us", but "we understood" does not have that same relationship to "he understood". Therefore, stress applies on two different cycles, with the first-cycle application of stress blocking syncope. Brame assumes that Syncope is a word-level (non-cyclic) rule, but that is not crucial and there is no direct evidence establishing that.
The interaction between Epenthesis and Stress is a dialect-variable example of cyclic vs. non-cyclic rule application. In many dialects, CC codas are broken up with an epenthetic vowel, usually [i], so that /katab-t/ → [katábit] "you wrote", /fihim-t/ → [fhímit] "you understood". Note that Stress must apply before Epenthesis, otherwise /katab-t/ → katabit → *kátabit. When the object clitic -na is added in this context, some dialects stress the epenthetic vowel, and some do not, thus you get [fhímitna] and [fhimítna] "you understood us". In the latter case, Epenthesis must be cyclic (so that the vowel can be stressed); in the latter case, it must be non-cyclic (post-cyclic) so that it is not inserted and therefore rendered stressable.
There are examples in English, which however depends on a less well motivated analysis set forth in The sound pattern of English: the Arabic example involves inflectional affixation and fairy non-controversial phonological rules.