The word you arrived with
Most people type the same word whatever their destination, because it is the one they were given by an employer, a university or a form. In ordinary use it has drifted into meaning "get my paper accepted overseas". In official use it means one narrow certificate issued under one treaty, and an apostille issued in Brighton for use in Laos is not accepted, so here the two meanings come apart.
That gap is not a pedantic point. The word decides which body you write to and which fee you pay, and the wrong word sends a perfectly good sheet into a queue that cannot finish it. Getting the vocabulary straight first is cheaper than getting it straight afterwards, when a file has already been posted and paid for.
What each name actually denotes
A certified copy is a fresh official sheet of an entry, made by the office that keeps the register. For births, marriages and deaths recorded in this district that office is Brighton & Hove Register Office, and the copy it issues stands on its own rather than depending on the sheet you already own.
The British certification is a different act by a different body. The Legalisation Office (Foreign, Commonwealth & Development Office) is the authority for it, and it charges £45 per document by standard post, or £40 per document on a next-day service for registered businesses. Consular legalisation, the third name, is an act of another state entirely and is what finishes a file on this road.
What goes wrong when the names are mixed
The commonest error is treating the three as stages of one product with one price. They are not. Each has its own body, its own charge and its own idea of what it is vouching for, and none of them will quietly perform another's job because your wording implied it. A request phrased in the wrong vocabulary is answered literally.
The second error is assuming that because a word is familiar it must be the right one. Knowing what you need for an apostille for Laos will not help you here, because that is not the certificate this route ends with. What helps is describing the errand by its outcome: a British paper that a body in Laos will act on.
The words that work on this route
Describe the sheet, then the stage, then the destination, in that order. "A certified copy of a birth record, certified by the British authority, then legalised for use abroad" names three real acts in the order they happen. Nobody handling your file has to guess what you meant, and no stage gets skipped because it had no name in your sentence.
It also keeps your expectations honest about the shape of the road. An apostille from United Kingdom to Laos is not the ending here, so any plan built on that single word is a plan with a missing stage in it. Naming the missing stage out loud is what turns a vague errand into a schedule you can actually work through.