Yes, both "davvero" and "proprio" translate to "really" in this context, but they have slightly different meanings.
- be an intensifier, as an synonym of "very" or "extremely" (ex: she's really nice)
- express that something is certain or true, as a synonym of "truly" (ex: he isn't really angry, he's just pretending)
One of the possible meanings of "proprio" is "truly", so it is a synonym of the second meaning of "really". Therefore:
- Ho davvero fame = I am really/very hungry
- Ho proprio fame = I am really/truly hungry
I put "I am really starving" which I thought (given "fame" can mean starvation) might convey the meaning of real hunger that they were trying to express. We would say I am really starving in English - not literally starving but to express that feeling of being very hungry. Are there any advanced Italian speakers able to comment please?
There are two reasons why. One, it's an idiomatic expression and the other if you break it down it makes sense. In an earlier lesson there was "ho fame" which is "I am hungry" or literal translation "I have hunger". Same principle here, "I have really hunger". -> "I am really hungry".
In combination with the response above this, here is why "sono" does not make sense.
As Briguy84 rightly points out, "Ho fame" traslates to "I have hunger".
If you switch everyting to English, knowing "sono" is "I am", then your translation of "Sono fame" would be, "I am hunger" , which is obviously incorrect.
Same goes for thirst and thirsty. Instead of "I am thirsty", Italians use "I have thirst".
Thus, Ho sete ("I have thirst"), and not "Sono sete" (which would be I am thirst.)
Other cases that need "have" instead of "am", are heat/ hot (I have heat; not I am hot), cold, fear/ afraid (I have fear; not I am afraid) and-- oddly enough-- the word "need" itself (I have need of, instead of "I am needing" or just "I need").
- Ho paura di (I have fear of)
- Abbiamo bisogno di (We have need of)