Well, this is a complex area and has alot of complications but in relation to your queries:
1. Is the onus on me or the Revenue to prove I was not in the country? How do you prove it? You are self assessed as a proprietory director so you have to tick the box on the form so initially you should have a basis for claiming you are non resident
2. What does 'Ordinary resident' mean, and how is this different to 'resident for tax purposes'? Ordinary resident means that you have been resident in Ireland for 3 consecutive years, so it takes 3 years to become non ordinarily resident. This is different to Resident as that is an annual test
3. If I live outside of Ireland for more than 6 months (183 days) then I am not liable for personal taxes, including PAYE, PRSI and USC. Is this correct? This is a very broad statement. There are 2 tests for residence the 183 test in a year or the 280 day test; if you are resident for more than 280 day in 2 years with a minimum of 30 in each year then you will be resident Normally where an employee leaves Ireland and takes up employment in another country they qualify for Split year relief ie they are deemed non resident from the date they leave. (does not apply to proprietary directors)
4. Does it make any difference that I will still be paid from an Irish company while I am abroad? Yes its fundamentally different for proprietary director
To answer your 2nd question it is clear that EO will not be granted to proprietary directors. You also need to consider the implications in the country you are going to.