OK, so now we have the following terms: State, nation, country, sovereign country.

For example in Germany one would say there are 16 states, but one wouldn't say that there are 16 countries. Never heard anyone claim that. So state seems to be different from country, at least here. Would people in the UK say that Scotland or Wales are "nations"? Or is at least that term very clearly only for units that could participate under a flag in lets say the olympics? (I assume here, that Scotland and Wales are not separately competing, but are under UK flag.)

"And by 'country' we mean a sovereign state that is a member of the UN in its own right" to quote the famous UK Quiz Show "Pointless".

Team GB enters the Olympics but as you may have noticed it was England who came Third in the World Cup recently, and likewise England play international rugby, including against Wales and Scotland.

The UK, crown dependencies and overseas territories compete as Great Britain in the Olympics rather than the UK. Northern Irish folks can compete for Ireland or GB.

As an English person I would say that Scotland and Wales are nations and countries; constituent countries of the country that is the UK.