The definitions don't sound the same at all.
If I immigrate to a country, then I am an immigrant. If I'm living outside my country, then I'm an expat. If I'm living outside of my country and migrated permanently to another country, then I am an expat and an immigrant. What the BBC is calling an expat is actually a migrant worker. Though technically they are a migrant and an expat.
You are only an immigrant if you intend on immigrating permanently.
You raise a good point - they are not mutually exclusive! Expat literally translates as 'Gone out from one's country' - but words are fluid and the more we can change them to be offensive, the better, I say.
Well, it comes from Latin "ex patria" and means from and/or out of the fatherland".