Here's a table that I think should boil things down to the most simple:
┃ type ┃ S │CE/CI/Z┃
┣━━━━━━━━━━━━╋━━━━━━━┿━━━━━━━┫ │ Where
┃ distinción ┃ /s/ │ /θ/ ┃ │ /s/ is like English S
┠────────────╂───────┼───────┨ │ as in [s]ing
┃ seseo ┃ /s/ │ /s/ ┃ │ /θ/ is like English TH
┠────────────╂───────┼───────┨ │ as in [th]istle
┃ ceceo ┃ /θ/ │ /θ/ ┃
The majority of Spanish speakers in Spain and Equatorial Guinea speak with distinción, that is, they distinguish the Z and soft C from the S. This doesn't mean, as it might sound to English ears that they speak better, just that they literally distinguish the two sounds. For someone from Madrid, caza sounds like what in English would be catha but casa instead is like cassa.
The overwhelming majority of Spanish speakers, though, including virtually all speakers from Latin America, are seseantes. That means they say pronounce words like caza and casa, or cocer and coser identically and when they pronounce them, the Z, soft C, and S sound like the English S. In other words, they pronounce caza and casa like someone from Spain would pronounce casa. (hence seseo, or using S)
A minority of Spanish speakers situated in South Spain, are ceceantes (there are also a few small pocket areas in Latin America that are too). They, like seseantes pronounce caza and casa the same. But the sound they use for the second consonant much more closely matches the one that someone from the North would use to pronounce caza. Technically, the sound is [s̪̟], but it's close enough that the term ceceo developed.
All of this came about because in the 13-16th century or so, there were a lot more sounds that were very similar and started getting confused and merged. As a result, even back then before all of the sibilant readjustment had completed they had multiple terms for pronunciations that didn't distinguish: seseo (as [s̺]), çeçeo (as [t͡s]) and zezeo (as [d͡z]].
I'm not entirely sure how someone pronounces the actual word ceceo outside of Spain, but my guess is that they pronounce it with the /θ/ sound, or the closest approximation they can so it's clear they're not saying seseo (and since the word is literally to talk about the sound used in speech, it's justifiable to use an otherwise foreign sound).