Of course there's a difference between "government" and "regime", otherwise we wouldn't need both words. Regime has more authoritarian and overbearing connotations which is why you will tend to see it more often used in relation to Trump, Thatcher, Erdogan etc and probably less with Major or Merkel. It will often be used to indicate that you are talking about an administration that is less tolerant of opposition or tries to bring in sweeping, possibly unlawful changes. The former describes May (she has called an election in order to try to reduce opposition within parliament as she has openly said) and Trump for example. It's also regularly used about Corbyn's leadership of the Labour party meaning that he doesn't brook opposition.
You don't get to define what a word means, the speaking and writing population does. And anyway Mark used the word totally correctly as defined by the definition you quoted!
But I realise I am getting ridiculously over-invested and it doesn't matter anyway, and I should get back to work.