Here's a thoughtco item on salutatio:
"In Ancient Rome, a Salutatio was the formal morning greeting of the Roman patron by his clients."
"The morning ritual was ... a fundamental part of Roman interactions between citizens of varying status."
Possibly the blind leading the blind here, but I'll give it a go. It's part of the word salutatio
formal morning call paid by client on patron/Emperor
salutationes - (noun, plural, accusative, feminine, 3rd declension)
The plurals all appears to pick up an extra consonant - the "n" - during declension. I'm not sure why.
So, in the end, we end up with a sentence that could be translated to something like "The clients make a formal morning call to the patrons."
clientes - the clients (nominative, plural)
faciunt - make ( verb, third person, plural)
Third declension nouns:
Looks to me like 'salutationes' means something like 'callings' or ' professional visits'.
My dictionary gives "salutatio -onis, f. greeting, salutation, a call, visit of ceremony."
Ancient Rome had a patron-client culture. And so in the context of 'the clients make visits of ceremony', it is implied that it is to patrons.
I don't see why "The clients are visiting the patrons" is given as incorrect.