This is an imperative / exclamation, - and it's what you would say to a person (e.g. your child) if they want the person to put on his belt.
As it is considered to be odd if you wanted the person to put on somebody else's belt (or clothes, glasses etc.) there is no need to be explicit on who's belt you want the person to put on!
I can think of cases where one may instruct a person to wear the belt that is provided, even though the belt may not necessarily be theirs. To me, this sentence should read "Put on the belt", if only because possession simply isn't indicated by the words in the sentence.
No it can mean seatbelt etc. too http://www.wordreference.com/iten/cintura http://context.reverso.net/traduction/italien-anglais/cintura
Indossare would be the verb to use for wearing something.
Yes. «-are» verbs when conjugated in the imperative mood switch the second-person and third-person endings; the «-ere» and «-ire» verbs keep the same endings in the «tu» form as in the present indicative. That is why «parlare» is «Parla!» for «tu» and «Parli!» for «Lei», but «mettere» is «Metti!» for «tu» and «Metta!» for «Lei». Similarly, «fuggire» is «Fuggi!» for «tu» and «Fugga!» for «Lei».
According to the following site, reflexive, direct and indirect pronouns are attached to the end of the verb for the “tu”, “noi” and “voi” forms, when the imperative is in the affirmative. https://onlineitalianclub.com/free-italian-exercises-and-resources/online-italian-course-upper-intermediate-b2/imperativo-parte-2/
DL gave only 'your' and not 'the' in the words to choose from yet the Italian sentence doesn't contain any indication of possessive. Surely the correct translation is 'Put on the belt'. Also it isn't good English to split 'put' and 'on' in this context. To put on e.g. put on a hat v to put a cup on the table.
This has been addressed in other comments; essentially, in Italian and other languages, since the verb is conjugated for «tu», it is implied that it is "your" belt. It is like saying «Lava le mani.» = "Wash your hands." Who else's hands would you wash? That is the idea. It is how Italians speak.