Juha, please don't worry if your question is not new.
The answer will anyway be new (not necessarily better ...) !,
although in this case not really complete, due to my work load.
Let me use my notation:
salutare_g : meaning in the context of /greetings/
salutare_c : meaning in the context of /cheers/
You use 'salutare_g' usually in indirect speech, like:
e1. salutami tuo marito. (greetings to your husband, say hello to your husband)
in direct speech it is usually restricted to exactly these expressions:
e2. ti saluto (plural: vi saluto)
e3. la saluto (plural: vi saluto)
When to use the previous examples?
(e1) is used when saying goodbye or more or less when the conversation is finishing.
(e2) and (e3) are used as 'good-bye', and sometimes are used to abruptly close
a conversation in order not to be overwhelmed by some feeling like anger or sadness,
or to avoid a possible verbal or non-verbal fight.
other common and 'mood-neutral' expressions are:
e4. ho fretta, scusate, vi saluto.
e5. {vado a salutare tutti, saluto tutti} e vado via. (suppose you are at a party)
In the old-fashioned postcards, there used to be a line such as:
e6. affettuosi saluti, e7. cari saluti, (In case you are too young and don't know what is /una cartolina/
just make a new question :) )
Regarding /salutare_c/, the common expressions are actually linked to the noun /salute/.
suppose you are drinking with someone who just recovered from an illness:
e8.1 alla tua salute.
Typical reply:
e8.2 ... e alla tua!
e9.1 salute! (cheers) literally: let's drink for your good health! (let's drink is not expressed)
e9.2 ... salute!
adesso devo proprio andare, ti devo salutare, ciao!
(here: proprio = really)