Kind of reminds me of when I was watching an international football game a little while ago and the commentator mentioned something about the 'Christian names' of the IRANIAN team....
To be fair though, here we are talking about a site written in English. In British, American, Irish, Canadian, etc.... culture first name-last name is the common format. Surely most Hungarians would know about the standard western order that is the norm in all of their neighbours and the English speaking world- hence when using an English language site they can expect this first name, last name format to apply.
I remember when I was in Japan and had to fill out some paperwork, the lady helping me out even directly translated the Japanese word for 'family name' into 'last name' for me, despite everything being the other way around in Japan. Its just one of the most common terms in English and should be looked at somewhat separately from its literal meaning.
If you use 'family name' instead of 'last name'....well even that isn't an innocent and culturally neutral term. Look to Iceland for instance where Bjorn Magnuson's son is Tor Bjornson.
Another thing I have wondered about here - what about Indonesians and other cultures where people just go by one name? There they can't just do a bit of logical translating and use some basic cultural awareness to enter their name properly....
I guess the answer is you just have to consider your target audience and try to cover for every eventuality there. If you expect things to be very international and it is very important you get someone's name right then perhaps including a little example with a famous name or two could help.