This word is commonly cited as a greeting (1, 2), and it’s obviously a compound of ὑπό+ἰαίνω, but it’s not in LSJ and it’s not in Perseus’ corpus. Is it a Byzantine invention? It’s not attested (as far as I can tell) in Modern Greek.
ὑγίαινε, perhaps? 
(“be well”, “farewell”, literally “be/stay healthy”)
stares blankly
That would be it, then. Many thanks.
Incidentally, do we know of any other greetings? Is there any known formality distinction?
Χαῖρε. 
See also here:
Colloquendi formulae: Tabula prima
Sprechen Sie Attisch?
OIKEIWN DIALOGWN BIBLION ELLHNISTI KAI RWMAISTI FAMILIARIUM COLLOQUIORUM LIBELLUS GRAECE ET LATINE
Unfortunately, my primary source for conversational Greek is Aristophanes, who, as far as formality distinction is concerned, has a very strong preference for the lower register. So I can’t help you too much. But if you ever found yourself in a situation where ὑγίαινε would seem too polite, you can use βάλλ’ εἰς κόρακας, “go to hell!” (lit. “throw (yourself) to the crows”). 
Joking aside, Aristophanes is a good source for conversational, streetwise Greek, I’m sure you can find some nicer and more polite stuff there as well, although you mostly learn curse words and how to insult other people. Some Plato’s dialogues could also be a place to look.
Looking at the second link you give, I wonder if some of them are not misleading. Someone can correct me if I’m wrong, but I would think that Ἀσπάζομαι would typically need an object, you’d say rather say Ἀσπάζομαι σε. I’m not sure about this though.
EDIT
I had a look at Sprechen Sie Attisch, linked by Bedwere above, and it also has Ἀσπάζομαι without an object. I suppose I was wrong then, you can use the word alone (For some reason, I have much confidence in old German books printed in Fraktur 
). If you can live with the German and the Fraktur, it looks like a very good resource to me. Thanks for that, Bedwere!
Welcome! I’m making a recording of the Greek in this thread.
υγιαινε, as Paul implies, is more often used in sign-off rather than greeting.
ασπαζεσθαι (occasionally in active form in letters) is transitive, requires an object, as Paul thought. Often used in the close of letters or as a PS: Give my greetings to XYZ, or X sends greetings.
The standard form of sign-off is ερρωσο (ερρωσθε), or later ερρωσθαι σε/υμας ευχομαι.