well, it all depends, in Spanish, at least in Mexico, we say, mi segundo nombre and also we use both last names (paterno y materno), here in the USA middle name can be either, un apellido o un segundo nombre
I'll try to clarify. A middle name is definitely a segundo nombre. In the English-speaking world, people only have one surname. Where they insist on using two, they hyphenate them to make them one. In Britain, this is usually an indication of aristocracy (or stupidity, or both), and it is called a "double-barrelled" surname (like a double-barrelled shotgun". So you occasionally get people called, for example, Henry Blenkinsop-Brown, or some such.
In the US, I have noted that many Spanish-speaking people who want to use both surnames use hyphenation too. So, for example, if the TV anchor man Joaquín López Dóriga were ever to go to live in the US, he would probably give his name as Joaquín López-Dóriga (the little hyphen makes all the difference). As for the surname/second name problem, many people in the English-speaking world have a surname for their second name (something I've never seen in Spanish). Indeed many also use a surname for their first name. Example: William Jefferson Clinton. There's no confusion about his appellido, and Willliam and Jefferson are his primer y segundo nombres.
And that's why I having so many troubles trying to use my two surnames in the United States, because they usually take my first surname as a middle name, and I had a lot of problems because of that, lost mail, name dissapeared in my university computer system, duplicity in a form for the visa. A total caos.
And the reverse is true also. I have a terrible time convincing folks here in México that my middle name is not my apellido paterno. My records get lost, my telephone directory listings are incorrect, etc., etc.
It works the other way round too, Julkio. As a naturalized Mexican I've had to adopt a second surname (my mother's last name), and -- because it comes last -- people at American Express in the States think it's my surname. It's a constant problem with banks, officials and so on. As you say, chaos. It has its advantages also, however. Mexico has an ID system called Clave Unica de Registro de Poblacion (CURP). I now have two CURPs, so I have a double identity, which might be useful for evading tax, I suppose!