The zipfile write() method supports an extra argument (arcname) which is the archive name to be stored in the zip file, so you would only need to change your code with:
from os.path import basename
...
zip.write(first_path, basename(first_path))
zip.write(second_path, basename(second_path))
zip.close()
When you have some spare time reading the documentation for zipfile will be helpful.