What is your direct clarification to my question?

Intention has two aspects:

The first is according to the people of the spiritual path (Ahl al-Suluk), which means that the deed must be purely for Allah, seeking His Countenance. This is the fundamental principle for a Muslim, and one should not succumb to the whispers of ostentation (riya').

The second is according to the jurists (Fuqaha'), where for purification, the intention of removing impurity (raf' al-hadath) or making prayer permissible (istibahat al-salat) is sufficient.