This is one of those FAA definitions that change depending on what you’re using the time for. Cross-countries fall into four groups. The first three groups are all contained in 61.1(b)(3).
Group 1: General Definition: A cross country flight is one in which you land at another airport that you didn’t accidentally bump into. There are no distance requirements.
Group 2: In order to “Count” for Most Certificates or Ratings: Same as the general definition, except at least one of the places where you land has to be more than 50 NM from where you started the flight. This applies to the private and commercial certificates, and the instrument rating.
Group 3: In order to “Count” for ATP: Same as for Most Certificates or Ratings, except you don’t have to land anywhere
Group 4: Apart from there are the “special cross countries” that are part of the experience requirement for certain certificates and ratings. One example is the private pilot certificate requirement for 150 total distance solo cross country with at least one 50 NM leg (61.109(a)(5)).
So, they’re all cross country. And they all can be logged from the time that you are a student pilot. The problem is keeping track of them so you can total the ones that “count” in any given situation. Most new pilots tend to log only Group 2 since those are the ones that they will have to total up in the near future. Some set up two columns right away (Group 1 counts for 135 experience purposes). The lack of a landing in Group 3 is a well-deserved tip of the hat to military pilots who will often fly some distance without landing.