What's the Name of this City, Anyway?

The choice marked (1) is the name in Polish, just as Copenhagen is København in Danish, Vienna is Wien in German, and Prague is Praha in Czech. We usually don't use these native names when talking about the cities in English unless we're pretentious, or perhaps sinking into inebriated nostalgia over some out-of-town escapade.

The choice marked (2) is what appears in airline schedules. It's neither fish nor fowl: not English, incorrect in Polish, but a useful convention in specialized contexts. 

The choice marked (3) is the English name for the city, in use for centuries* and still regarded as standard in serious works.

* I remember how, back in grad school, I came across a character called "Count Lubomirski of Cracow" in, I believe, Thomas Shadwell's 1676 London stage hit The Virtuoso (if memory serves; it's been a while). Little did I know at the time that I would later meet the current incarnation of Count Lubomirski in Cracow and, ca. 2000-2002, regularly run into him because we both shopped at the same grocery store on ulica Dluga. We still exchange bows when we pass on the street, although I'm sure he's long forgotten who I am. 

Note 1: Poland and its larger cities have been a part of the European mainstream at least since the days when Cracow was the capital of the continent's largest country and Warsaw was an important provincial center. Like Rome, Moscow, Belgrade, and Cairo, they have earned the right to be called by their English names. Poland has no need to resort to the resentful nationalism, ever-so-seventies and ever-so-non-aligned-movement, that led some countries in what used to be called "the underdeveloped world" to insist that everyone else use their native geographical designators. 

Note 2: Each of the three choices listed above is correct in the proper context. Mutant forms like "Crakow," "Kracow," etc. are wrong in any context.