A good question. “Full name” usually means all the names. In the US, we would probably say something like Her first name is “Elizabeth” but she goes by “Liz”. Her given name is “Elizabeth” but she likes to be called “Liz”. “Pet name” wouldn’t work, since it would only be used by a limited number of people. “Nickname” is another possibility but doesn’t sound quite right for a name like “Liz” which is very common (and also a given name). If you have to use a label, “short name” sounds good to me.

We would normally say 'Liz is short for Elizabeth' or 'Liz is a short form of Elizabeth'. A pet name is a name given to express fondness and is only used by certain people, rather than for general use. It may be a short form of the official name, but it might be an entirely unrelated nickname. For example Elizabeth might be called Liz by nearly everybody, the short form might be her normal name, but her husband might call her 'Miss Sunshine' that would be a pet name