I agree a first fast should be done with care but I don't think a Water Fast should be avoided because it "won't take [toxins] out as quickly as possible". In fact, it is because a Water Fast DOES take toxins out quickly and potentially overwhelmingly, that one might consider a juice fast or what you mentioned prior instead of a Water Fast to begin with.
Though again, a water fast may be appropriate as a first fast, but according to the state of toxicity and health of the practitioner, maybe initially for just a short duration.