Within the Schengen zone it's like that second paragraph, even for cities in different countries.

(Whilst the UK does have high speed trains to other cities in Europe from St. Pancras in London, even back before Brexit you still had to go through passport control to board so it does add a bit more overhead to the train trip, plus those trains are normally pretty expensive if you don't book a month or two in advance)

I remember this one time when I went to ski in Austria and to get there I had to catch a train from Innsbruck to the village nearest the resort and as it so happens that was a train that had departed from somewhere in Germany, was passing through Austria and was going to end somewhere in Switzerland. Normal interurban train (not even a high speed train) making its way through the Alpine valleys in Austria, that just happened to stop in cities in 3 different countries.