First: pure and purifying are not identical terms. Water may be pure while its purifying capacity remains in doubt.
Second: disliked does not entail impure; cat and bird-of-prey su’r illustrate this.
Third: juristic disagreements require attention to the rationale of the proof. The vessel-washing instruction for dogs leads to different conclusions for Malik and Hanafi jurists.
The next independent chapter is tayammum.