I would say "visiting a country" requires at least seeing one meaningful landmark or some noteworthy place of interest or "thing-to-do".

1) Definitely a layover in an airport DOES NOT COUNT as visiting the country. Neither would an overnight stay in an airport hotel.

2) I also believe that what is visited in the country should be of relative importance and also be representative of that country.. For instance, if visiting the Maldives, you'd probably want to hang out on a beach, no need to spend time in the capital city... But staying a few days in Puerto Rico or Hawai DOES NOT COUNT as visiting the USA... Applying this logic to Russia, a visit to Volgograd would not, in my opinion, count as a "PROPER" visit of Russia because Moscow and St Petersburg hold so much importance. But in the USA, importance is spread out between cities such a NY, LA, Chicago, DC, Miamiii and Natural Wonders (i.e. Niagara, Grand Canyon, Monument Valley) and one cannot be expected to visit everything, but at least 1 or 2 of these...

3) Basically, for someone to be able to say they have "visited a country X", they should be able to aanswer the following question: "Did you enjoy Country X?" or "What did you think of Country X?"... Again, if you transit through Frankfurt, you would not have the credibility and credentials to be able to comment on German culture or the beauty of German cities/nature...