Why was the King of Britain the Emperor of India, and not the Emperor of all the other colonies? Why did this distinction only come in in the late 19th Century?
Why was Russia an Empire despite having no colonies? (Georgia/Armenia/Khazakhstan/ et al count as colonies in this instance?)
Why could Napoleon proclaim the First French Empire despite France losing many colonies prior to this?
I'm assuming it has to do with rule over different national and cultural groups, but then couldn't this apply to most Kingdoms (eg Emperor of the English, Scots, Welsh and Irish?) Why was the Emperor of Austria not the Emperor of Hungary? (I realise there are issues with the dual-monarchy and HRE in this example)
In the British example the answer is quite simple. The idea that the Crown in India was really an empire had been floating around for a while but the main reason was that Victoria herself wanted it. Firstly because she, and everyone else, could see that the monarchy was a declining power and the boot in prestige that came from the assumption of an imperial title might be useful. Secondly, she was annoyed that her daughter Victoria would be becoming empress of Germany. Technically as empress she would be one rank higher than her mother and this rankled Victoria to no end. Now, Disraeli the prime minister made his career partially by being really popular with Victoria and in his own words "laying the flattery on with a trowel". While he wasn't hugely enthused he felt that, after years of pressure from her that, he could prevaricate no more and got her the title. He did however kill her idea to make her the empress of Britain and Ireland as well. The empress of India was created by the Royal Titles Act of 1876.
There have been a few threads about the fluctuating use of "king" (or "queen") versus "emperor/empress" in the past.
Tl;dr an "empire" is usually thought of as a multinational/multiracial/multiethnic group of countries/principalities/soverign states, depending on how that was defined at the time, so you have the king of Hungary but the emperor of Austria-Hungary.
But there are a ton of exceptions, English is weird and rulers call themselves what they want.
Here are some relevant threads:
https://www.reddit.com/r/AskHistorians/comments/3ptbu7/why_isnt_the_queen_of_england_titled_empress/
https://www.reddit.com/r/AskHistorians/comments/2ygnra/what_is_the_difference_between_a_king_and_an/