Oh, that makes me feel less stupid, knotty. 
Becuase I didn’t understand that part either! 
But I felt it doesn’t much matter because their conclusion was that this all didn’t apply to the common nāmarūpa in Buddhism, anyway, where it refers to the individual. I also got the strong sense that they were skeptical whether it applied even to Brahmanism, when they say “may I say, actually [nāma doesn’t mean ‘designation’]”.
I understood there to be three ideas presented:
- nāmarūpa means some sort of designation/recognition process (which they explain with these models that we don’t understand)
- nāmarūpa means external objects with intrinsic “names and forms”
- nāmarūpa means the "internal"being with its intrinsic “name and form”
I felt Olalde disregarded the first in general for both Brahmanism and Buddhism, and the second for the common factor of nāmarūpa of Buddhism also. Which leaves us with the third for 99% of instances of the word in the Pali canon.
I may oversimplify it or misunderstant the nuances. Either way, this is my own personal conclusion, whether it was Olalde’s or not. Because it aligns with what I’ve read elsewhere. For example, Olivelle’s introduction to his translations of the Early Upanishads (recommended) touch upon it too, saying: "The essence of a thing [or person] was expressed in its name and its visible appearance (nāmarūpa)”. He doesn’t speak about recognation at all, if I remember correctly. Neither do other scholars of those texts I’ve read, like Keith.
I think Olalde is siding with those more “traditional” interpretations like that of Olivelle (nr 2 & 3) and arguing against the interpretation they say arose in “the last decades” (nr 1).
In the last decades, however, nāmarūpa has attracted the attention of scholars of Buddhism who turn back to the “older” usage in order to elaborate alternative interpretations that aim to grasp the “original” meaning of the Buddhist term. […] They understand “name” as “designation” and neglect the fact that italso (or may I say actually) means “proper name;” in this manner they understand name as “naming” and assume that it always encompasses conceptualisation.
Now, it’s a bit scholarleze, but the quote marks around “original” and the “may I say actually” to me indicate that they don’t think this idea is actually more original and correct.
It also should be noted that the Buddha did not fully adopt the cultural concept of nāmarūpa. People thought their names were part of their selves, but the Buddha didn’t agree with that idea of course. Still, he used nāmarūpa pragmatically in most his teachings on dependent origination, where it refers to the being. This much is quite clear from passages where nāmarūpa is conceived in the womb. But at times he also criticized the idea that the name is a part of the self. Partly this is reflected in him redefining nāma to be feelings, perception, contact, and so forth. That is, he’s saying it’s not really what people thought it to be. Part of this is perhaps also the cryptic passage on nāmakāya and “designation contact” in DN15.
But some people have gone too far with these critiques of ideas of nāma, imo, and fall into this “designation” idea. (Like in the other thread where we met.) Others, like the Abhidhamma, miss the cultural idea entirely and partly therefore redifined nāma to be the four non-consciousness aggregates.
So you see, there’s multiple layers culturally and historically, and that makes nāmarūpa such a tricky term, and why it is difficult to translate. For example, if you translate “mind and body”, like some do, then it is pretty self-explanatory for the standard way it is used in Dependent Arising, but then you totally miss out on the critique of the concept of one’s name being part of one’s self.
So translators choose what to highlight. Or, out of desperation resort to literalism. 
And then there’s more tangentially connected ideas with speech and the divine power of sounds in Brahmanism. This is not my area of knowledge to explain. Olivelle I mentioned earlier is a good read. Not too much details, explains the gist.
Most important conclusion for Dependent Arisng, though, is Olalde’s note that it refers to the being with its name and body (or shape).
That is how I understand it. Let me know if you interpret the paper differently.
PS. Let’s not get started on rūpa 
Because that also has various nuances. If you imagine a color, for example, it is also called a rūpa, even if it exists only in the mind. This all has to do with the concept of material being different back then too, not being limited to atoms which were only discovered recently, in the scope of time. (Which, of course, also is challenged by more modern science.)