IMHO - The scond equations is incorrect! - because the second dr in the first
equation is 0. Actually performing \int involves a path, not just a change in phi
for reasons independent of dr.
You can change the energy of q by changing potential (by moving other charges),
but it's not work done on q itself, V has changed because of energy input
elsewhere in the q1-field-q2 configuration.
Maybe I'm seeing this totally wrong - I'd appreciate a clarification if I am.