The detail that best shows Clara’s purpose for writing "A Story of the Red Cross" is:
"I have dedicated this little volume to the people with whom, and for whom, have gone the willing labors of twenty-five years—initial labors, untried methods, and object lessons."
This statement indicates that Clara is writing to honor and acknowledge the work done alongside those involved in the Red Cross, highlighting her dedication and the collaborative nature of their efforts in humanitarian relief.