"Second" used to be the communist bloc, led by USSR. USSR was disbanded about 30 years ago, but the idea of numbered "worlds" persist.

(I'd assume that Russia and China are first-world countries now.)

The late Hans Rosling had a good update to the old conceptions of 1st, 2nd, and 3rd World. His take was to classify based on income levels on a log scale (well, 'log-ish', anyway). Hence the classifications of Level 1 (<$2/day), Level 2 (<$8/day), Level 3 (<$32/day), and Level 4 (>$32/day). This is my preferred way of mentally chunking the world and her people. Mostly because a lot of people are moving on up!

https://www.gapminder.org/fw/income-levels/

If you've not played around with the stats on gapminder before, go grad a cup of coffee and have some fun! The visuals are good and the interface is one where 4-D data is well presented. In messing around with it, you can see that the world is very much on an upward trajectory, but also has a long way to go.

https://www.gapminder.org/tools/#$chart-type=bubbles

there are no first world countries among former third world countries, only first world cities

Well, first world regions too, sometimes.