Different situations have different definitions of 'resident'.
For the foreign-earned income exclusion link you posted, you need to have resided in the UK for at least 183 days of the tax year to be considered a UK resident. The UK tax year runs from April 6th to April 5th, so by the time you get to the end of this current tax year, you will be considered a UK resident for tax purposes, as you will have spent more than 6 months of the tax year in the UK.
For travel insurance, you will have to look at each travel insurance company's definition of being a 'UK resident' to determine whether or not you qualify for their insurance:
- some companies require you to hold ILR (which means you've resided in the UK for 5 years and have permanent residence
- some simply require you to be 'ordinarily resident' in the UK (your main residence is in the UK)
- some require you to have been resident in the UK for 12 months or more, some 6 months or more, and some require 3 months or more.