There have been a lot of attempts to describe the difference between phonology and morphophonology, terms that really go back to Trubetzkoy's Principles of Phonology. However, the phonological theory that Trubetzkoy was working with had actually been started in the 1870s by Baudouin de Courtenay, who first coined the word "fonema" (phoneme) and kicked off the synchronic study of languages. Baudouin was the first linguist to develop the concept of sound alternations, which became the basis for Trubetzkoy's distinction between phonology and morphophonology. Baudouin called Trubetzkoy's "phonology" "physiophonetics", and his "morphophonology" "psychophonetics". However, Baudouin was not a structuralist in Trubetzkoy's or Saussure's sense of that term His physiophonetic and psychophonetic alternations had very distinct psychological functions, whereas structuralists tended to downplay psychological function or to view linguistic systems as primarily social in nature--i.e. related more to Saussure's perspective on language.
For Baudouin, physiophonetic alternations were distinct phonetic pairs that represented a single underlying phoneme. The phonetic alternants could be purely allophonic or involve complete phonetic neutralization. So Baudouin consider the s/z alternation in, say, Polish mróz [mrus]/mroza [mroza] ("frost"/"of frost") to be simply /z/, since Polish devoices final obstruents. That is, the nominative form was /mruz/ for him. He believed that Poles (like himself) believed that they were pronouncing a final [z] as an [ s] sound. OTOH, Baudouin held that the u/o alternation involved two phonemes /u/ and /o/, because Polish has no physical constraint against pronouncing either of those two sounds in that position in the stem of the Polish noun.
So, turning to Trubetzkoy, there are some significant lessons to draw from these historical facts. One is that Trubetzkoy includes both allophonic variation and phonemic neutralization as part of his theory of "phonology". Unlike Baudouin, he did not see final voiced obstruents in Polish as phonemes. Rather, he saw them as archiphonemes that were realized phonemically as voiceless in final position and voiced elsewhere. Trubetzkoy, a Russian linguist trained in Moscow (not Baudouin's St Petersburg), was simply modifying Baudouin de Courtenay's take on physiophonetic alternations. Another lesson to take away is that Trubetzkoy explicitly excluded morphophonology from his theory of phonology, preferring to treat it as being a distinct component of grammatical description. That is because Trubetzkoy took it for granted that Baudouin's physiophonetic/psychophonetic alternational dichotomy represented distinct linguistic phenomena. He bought into Baudouin's dichotomy, but he modified it to make sense within his "structuralist" framework. Archiphonemics and morphophonemics belonged to distinct areas of the grammar. This concept of a fundamental dichotomy essentially evaporated in other structuralist approaches to phonemics and was completely obliterated by Chomsky and Halle in the 1960s.
Now, regarding the problem mentioned in the OP, here is likely the way Baudouin would have analyzed it. The process in question is sometimes referred to as "assibilation", where an affricate like t͡s reduces phonologically to [ s]. Baudouin would likely have analyzed the underlying word as the phonemic sequence /nt͡sunu/, where the psychologically real t͡s phoneme corresponded to the [ s] alternant of that phoneme in pronunciation. Trubetzkoy would have used an archiphoneme to describe the same alternation.
Finally, it is worth noting that Baudouin's theory of phonetic alternations evolved into three distinct Russian schools of phonology: the Leningrad School, the Moscow School, and the Prague School. All three schools would have come up with different analyses of the t͡s problem, but the Moscow School would have retained Baudouin's analysis most faithfully. The Leningrad School did not allow for phonemic neutralization, so they shifted the boundary of phonology/morphophonology, lacking the "compromise" of Trubetzkoy's archiphonemes. Trubetzkoy retained Baudouin's original physiophonetic/psychophonetic dichotomy, but he renamed it phonology/morphophonology and added the concept of archiphonemes in cases of phonemic neutralization.
