[Facts] Re: What's the etymology of Salutius and Salutia?
My Latin is rusty; I hope this helps!
I haven't found a Latin name like Salutius; Salutia of course is just the female form. But the origin would be the word 'salus' = 'health'; Latin being an inflected language, the word endings of a noun change according to where it is in the sentence, and the genitive case of salus is salutis, not saultis! All the other cases contain salut- and then their own ending.
There was a goddess called Salus who was worshipped in Rome.
The verb 'salutare' means to greet someone; our word 'salute' comes from it. And the verb 'salvere' means 'to be well'.
Salve! is a greeting for one friend; Salvete! for more than one. (Ave! is a respectful greeting; think Ave Maria.)
Salutus could I suppose have been used as a given name. And putting the -i- in there could indicate a younger relative of Salutus: father Salutus and son Salutius, perhaps. Along the lines of Iulus (original) and Iulius (his descendents).
So the meanings could range from 'Be well!', a greeting used by my Jewish friends, to 'healthgiving', like medicines, to 'pay respects to' someone, which means that just about all the meanings you've listed would be appropriate, according to context of course. And, the original meaning would have been a physical condition: bodily health. Salvation, the health of the soul I suppose, and a figurative meaning, came later.