The more questions you ask the more I am thinking you'll make a fine instructor.
Starting from the get go, let me put your mind at ease and point you in the direction of definitions. The term cross country is a vast one depending on what it is we are discussing, in your case it is very specific.
Regardless of what is it that we are discussing the definition,accepted worlwide, of a cross country is taking the aircraft out of an airport and repositioning it at another airport. regardless of distance.
Yet as you pointed out if we are looking at acquiring ratings and certificates 14 CFR 61.1 will define what exactly is a cross country so that it is meaningfull in its experience and truthfully claims to have imprinted knowledge into the airman claiming such experience.
For most certificates that would be the 50 nm point which is designed so that with a light training aircraft, upon reaching altitude, you cannot see the arrival, hence you will need to use and demonstrate a minimum knowledge in navigation (whatever that method might be).
Yet, as a flight instructor you are offered to endorse a student for repeated cross countries which are less than 50 nm in length. So your question as I understand it would really be what is, if any the purpose of such flight?
Here is your answer:
Some flight schools, determine that students cannot proceed with repeated touch and goes at the home base airports. This reasoning could be a resultant of noise abatement, high heavy traffic, etcetera. If that student is to go practice his skills solo, an endorsement might be necessary for him to go beyond 25nm , but less than 50nm, to an airport which offers no noise abatement, and lighter traffic. That actuallly was the case at the last flight school at which I worked. So the regulations provide you with a loophole for a student solo pilot to go beyond the realm of the 25nm he is initially limited to, providing of course that you have endorsed him.
As far as the requirements for such endorsement they are the same as for any cross country flight endorsement.
One thing to point out here the feds are using the word cross country in its true meaning, the one I gave you earlier: the repositioning of an aircraft from one airport to another, regardless of distance. Alas, we both understand that this student is not going to be logging any X-C time on this flight!