This is a purely terminological confusion. The diagram is supposed to convey that in some specific cases, a head is called a nucleus. But in the "Head" article, the word "nucleus" is used in a different sense.
The terminology used in the diagram is that of The Cambridge Grammar of the English Language (2002). The user who added this diagram back in 2009 is a linguist who works in the GGEL framework. So an explanation of the diagram has to be specific to that framework. The other part of the question is easy: for the other Wikipedian, any head is also called a nucleus, so I'm gonna ignore that and focus on CGEL.
In this framework, there are three specific cases where heads have different names: "predicate", "predicator" and "nucleus". These names are optional: you can just call them heads. The head of a clause is called predicate, and the lexical head of the predicate is called predicator.
Now, a nucleus is another name of a head that governs a specific kind of dependent: an extranuclear one. When a clause goes from the regular English SVO order to a different order, we say that something is moved to an extranuclear position, and what's left is a nucleus that still conforms to the SVO order, even if some of its constituents are now gaps (or "traces"). This is useful for the purposes of classifying different non-canonical constructions: it highlights a firm foundation — a canonical clause with a fixed word order — from which specific non-canonical construction deviate in various ways.
If the extracted bit goes before the nucleus, it's called a prenucleus. If it goes after, it's a postnucleus. We use indices like "(i)" to show that a particular gap within the nucleus refers to a particular extranuclear constituent.
Examples with prenuclei:
- Subject-auxiliary inversion. I am here changes to Am(i) I ___(i) here?
Am is extracted to the prenucleus position; I ___ here is a
- Relative constructions. Clause I met this person to NP the person
who(m)(i) I met ___(i). The extracted wh-phrase is a prenucleus; I met ___ is a nucleus.
- Preposing. I can prove these theorems changes to These
theorems(i) I can prove ___(i). These theorems is an extracted NP; I can prove ___ is a nucleus.
- Open interrogative clauses. These have two extractions and two gaps. He chose X changes to What(i)
did(j) he ___(j) choose ___(i)? The nucleus is he ___ choose ___, and two constituents are extracted to prenuclear positions.
Examples with postnuclei:
- Postposing due to weight. Usually, a direct object will immediately follow its head verb. But if the object is relatively long and complex, or "heavy", it gets relocated to the postnuclear position at the end. For example: We found ___(i) in most countries [a fundamental conflict between the elite culture and the rest of the population](i). Here, the bracketed part is a heavy direct object that is separated from the verb by an adjunct.
- It-clefts. They couldn't prove the latter theorem changes to It's the latter theorem that they couldn't prove. This construction is a bit different: nothing is extracted (apart from the relative extraction), but the words are still rearranged in a specific way. The original clause is split in two, with the second one being a relative clause containing presupposed material. The relative clause that they couldn't prove is a postnucleus here.
Example with both:
- Subject-dependent inversion. The dog ran away changes to
Away(i) ___(j) ran ___(i) the dog(j). The complement away is preposed, and the
subject the dog is postposed. So the nucleus is just ___ ran ___.