Ciao!

come si dice in italiano "berthing" anche "unberthing" ?

See a translation

As a noun berthing can be translated into italian like 'ormeggio', 'ancoraggio', 'attracco' while unberthing would be 'disormeggio', 'disancoraggio'. As a gerund, berthing would be translated into Italian like 'ormeggiando', 'ancorando', 'attraccando', and likewise unberthing would be 'disormeggiando', 'disancorando'. Please note that, as unpredictably as it seems and differently from 'attraccare', the verb 'disattraccare' isn't listed in the italian dictionary, and therefore isn't allowed to be used, so that 'disattracco' and 'disattraccando' aren't allowed as well.

Sorry for the mess, but I've just found this way to answer your latest question:

I guess that in English DISANCORAGGIO is expressed with 'weighing anchor', or '[(taking/pulling) up/lifting] the anchor', and even other verbs not including 'anchor', like 'loose {for sea}' or 'unfasten' are preferred, although I'm not completely sure about that, so this would be matter for a question in the English (US) section of this blog. I think ANCORANDO = anchoring is correct. But if I were you, I wouldn't worry, because any English and Italian native speaker will understand you if you say anyone of those terms in their respective languages.
No, tokkyu_nibbio, my job has no relationship with the sea, I'm an IT technician. So long, tokkyu.