I was fired by my foreign employer in the end of 2018 and the severance pay was paid in the middle of January 2019. In this case, should I report my severance pay as part of my foreign earned income and for which year?
As long as you were in a foreign country when you performed the services that gave rise to the income, it is foreign earned income. This would include severance pay.
The income is applied to the year in which you actually earned it, regardless of when you received it.
Therefore you would report the income on your 2018 return, and you would include the severance pay.
https://www.irs.gov/newsroom/foreign-earned-income-exclusion-2016
That would not be correct. The income is still reported in the year it is received. However, the test for whether it is excludable is based on the year it was earned.
IRS regulations deem severance to be “earned” income and to be earned when paid. (If your contract or the laws in a foreign country specify that you are entitled to so much pay per year of employment, that does not mean the severance was “earned” years ago, at least under US tax law.)