Did any British Monarch ever claim to be an emperor?
I've written about this before. c-p is below.
This question comes up from time to time, and here are a couple of previous posts that may be of interest:
https://www.reddit.com/r/AskHistorians/comments/44wszk/the_fluctuating_use_of_emperor_and_king/
https://www.reddit.com/r/AskHistorians/comments/3ptbu7/why_isnt_the_queen_of_england_titled_empress/
To summarize from that last one: We have in fact in the past referred to British monarchs as emperors or empresses, starting with Victoria in 1876 and ending with George VI in 1948. They were styled Empress/Emperor of India, to go along with their other royal titles and styles. There's a whole table of the monarchical styles from the creation of the kingdom of Great Britain in 1707 through the present day in the link, which may be of interest to you.
My guess is that Britain was never formally an empire but rather a kingdom, the United Kingdom. However they did have overseas colonies which leads me to the conclusion that the UK is a kingdom that had an empire.
You are on the right track here -- in English, an emperor is usually someone that rules over a collection of distinct countries or ethnic groups which have united to form an imperial organization. In the case of Great Britain, the kingdom itself had overseas colonies/holdings governed from the home office, but not with full membership in the nation-state (ex: the American colonies didn't have votes in Parliament, etc.) The monarch might have personally been the ruler of those countries regardless of whether or not they were part of the nation-state of Great Britain (ex: the Hanoverian kings of both Britain and Hanover, who were legally distinct people who happened to inhabit the same body.) And there are examples of monarchs of nations that are independent of Great Britan but still have the same royal family (ex: Elizabeth II is the queen of the UK, Canada, Australia and New Zealand, as well as the Commonwealth nations, including countries that have become independent of Britain since she became queen: Jamaica, Barbados, the Bahamas, Grenada, Papua New Guinea, Solomon Islands, Tuvalu, Saint Lucia, Saint Vincent and the Grenadines, Belize, Antigua and Barbuda, and Saint Kitts and Nevis.)
But there are a whole host of reasons why we call someone an emperor and not the other person, especially when translations come into play -- English is weird.
(Also, I copy-pasted the list of those countries from Wiki, I can never remember it completely.)
Also, here are some past threads:
https://www.reddit.com/r/AskHistorians/comments/2ygnra/what_is_the_difference_between_a_king_and_an/