Your London friend is sadly misinformed. Here iss the British English only corpus from Google NGrams:
So Brits definitely stick with surname, but in the United States of America, there has been a noticeable shift towards last name in recent decades:
Probably most Anglophones have one middle name, but there are plenty of people who only have a first name and a surname. And many people have two or more of what I've never heard called anything other than middle names (first and middle names collectively are called forenames).
Over recent decades, people increasingly tend to avoid the religious implications of christian name. Somewhat surprisingly to me, it appears first name is more favoured in the United States of America than Britain, but I suppose that is more by analogy with last name than because of secularist leanings.
Forms such as given, birth, married name etc., are relatively uncommon, in that they're only normally used in contexts where they're needed to disambiguate. Usually, given name(s) are forename(s) your parents gave you at birth, if these aren't what you call yourself in later life (for whatever reason). That can also apply to birth name, but this is also used to contrast with the married name (most women still adopt their husband's surname, which may also be indicated by Mrs. Smith, née Jones).
As the charts show, second name is quite rare compared to surname (and last name in the US). Here is a link to a chart showing it is equally rare compared to first name, Christian name, and middle name. Partly because it's uncommon, it has no established unambiguous definition. Here is a typical online entry:
second name (second names, plural) Someone's second name is their family name, or the name that comes after their first name and before their family name.
(where family name is just a somewhat less common alternative to surname).
Collins Cobuild English Dictionary for Advanced Learners (4th edition) © 2003
And a "real" dictionary (such as the Oxford English Dictionary) does not even bother to define such a vague collocation. As various comments indicate, it could also mean a nickname, stagename, alias, nom-de-plume, etc. In fact, to different people in different times and places, second name can mean just about any name or part of a name that's not the first name. It is no more meaningful than other name outside of context (personally I think it sounds a bit childish and uneducated, but that's just me! :).
The question of "where you are from" is complicated for many English-speaking people because they live in North America, where there are historical and political considerations about where you are from.
I consider myself to be "from" Canada yet my paternal grandfather was born in Britain somewhere (probably Wales). I don't consider myself "from Wales". My maternal grandparents are from Quebec; their families had lived there for several generations. At what point do they stop being "from France"?
Yet I can't call myself a "Native Canadian" without confusing some people because there are other peoples, the First Nations (or Indians) who are more commonly referred to as "Native Canadians".
Given that so many people are immigrants, or relatively recently descended from immigrants, it usually takes several questions to identify people's family backgrounds. And that is maybe one of the better terms to use: Family Background. You can ask what someone's family background is and they might have a quick answer, such as "China" or "Ireland". I still don't have a good answer for that either because it's not a simple question.
Generally there is no such thing as an official "origin" for your family that would be recorded on ID cards or other official papers, thus there aren't really any good single words or set phrases to label these origins.
Finally, your suggestion of hails from only works if the person or group in question is recently from the place you're naming. You hail from the place you were born or grew up. You could describe your family as hailing from some place, but if you're trying to say that generations ago your family "hailed from" New York, you'd have to use extra words to indicate that. "My family hails from New York" would be most likely interpreted as your parents and yourself being from New York. If your parents had never lived in New York, I'd suggest you should specify that: "My family originally hailed from New York." Hails from is not a formal term.
Best Answer
While one can't be BORN by one's mother, all of us are BORNE by or of our mothers.
One could, however, say that one was sired or begot by their father and BORNE by their mother, both of whom are their progenitors.
Your examples:
1) I was born John Smith by Jane Doe and Dave Smith; 2) I was born John Smith by Jane Doe to Dave Smith – neither example is grammatically correct.
Correct possibilities:
• I was BORN John Smith (meaning having certain qualities or characteristics from the time of your birth, in this case, the name John Smith), TO or OF Jane Doe and Dave Smith.
• I, John Smith, was BORNE [meaning carried or given birth to] BY or OF my mother, Jane Doe.
(linked definitions courtesy of Merriam-Webster online)