Remember when we had that topic of what we sound like in English? I did think that something like this was probably the case back then, but I didn't want to sound overly critical by saying it
The thing is that I think it's fairly easy to say that English is not our native tongue simply by the fact that we were not born to it -- it's not our mother tongue, and we spend the most formative years of our lives not speaking or even hearing it. It's only layered on top of the language of our birth, something we learn later. You say you think in English, and I will admit I sometimes to that too (because I spend too much time on the internet), but it's your English, already internalized through your accent.
That is what I meant when I said that the British of my schooling and American of my culture intake is still layered on top of my own original Swedish accent. I don't sound too thickly accented, I don't think, and many years ago I got a Cambridge "certificate of English speaking" and did very well, so I know that I (at least at that time) is not too bad at English. But my Swedish accent still comes through when I speak -- unless I deliberately make a conscious effort to speak English with some kind of accent, it will always default back to an English were the tone, stress, and speed is influenced by my Swedish. Because even though I on a logical level know that that isn't what English is like, that is still what body has internalized English as.
Also remember that hearing/understanding and speaking languages are too separate skills. I have a Catalan uncle who, after years of living with his Swedish wife can follow along when people speak Swedish and almost completely understand it, but he himself
speaks only the barest of words. To learn to speak a language one must practice speaking it, not just hearing other people talk in it.
It's likely the case that you speak with some amount of accent, but not as much as people in your country generally do. Then, because you have less of an accent than they are used to, you come off as speaking better English to your family and friends -- for them it is how you differ from the general way of speaking there that matter, it's the absence of strong [your language] dialect that is remarkable, and thus what they notice. But for somebody from Italy, who is conditioned to Italian and Italian accent, it is how you differ from English that is noticeable, so they can quickly tell by the accent you have that it isn't mainline English.
The reason they guessed Indian is probably because they're not at all familiar with of the dialect of your language (or languages in your general region) sounds in English, so they just guessed some accent they were more familiar with that it sounds remotely like in their ears.
I also want to end this by saying that speaking with an accent is not something bad. I used to be ashamed of the mere concept of speaking with an accent and want to do away with in entirely, but over time I've changed my mind. My accent is as much a part of who I am as anything else, something that informs others about me without me even having to expressly tell them about myself and my history. Accents are beautiful, my friend.
Or maybe I've just grown less inclined to build my self-image on how well I do things like speak English and the like.