What separates "rich" developing nations from "poor" developed nations? What separates relatively high income developing nations, such as Argentina, from relatively low income developed nations, such as Portugal?

If you look at things like HDI numbers, Argentina and Chile are classified as very high and even rank above Portugal. If you look at per capita GDP then Argentina, Chile and Portugal are all pretty close and fall between 22K and 27K. 

Despite this, Argentina and Chile are generally considered developing countries despite being on par with or exceeding some countries that are generally considered developed. 

This makes me wonder, what exactly separates developed countries from developing countries?

When comparing the United Kingdom and Uganda the answer is obvious, however it isn't as obvious (at least not to me) when comparing countries like Portugal and Spain to Chile and Argentina.
All country classifications are conventions, which depend on the rule used by the particular institution. There is no hard (ontological) rule on any classification. There is no even consensus on the definition of categories. Advanced/rich/developed/industrialized economies are sometimes used interchangeably, but who is in each of them depend on who is talking. Let see an example:

International Labor Organisation (ILO) 



(I had to cut it because it is too long. See the full list in page 67 of this report).

Citigroup



(Report here).

Conclusion

As you can see, ILO puts Czech Republic, Greece, Argentina, and Chile as developed countries, whereas Citi puts them as emerging countries (Argentina as emerging can be seen in other reports). (ILO even distinguishes between emerging and developing).

This shows that there is no hard-coded rule about classification of countries. It all depends on the institution.