Check the definition of probabilistic classifier. As far as I see it, we define it to have scores in [0,1]. But IMO this is not enough and a true prob classifier should have probabilistic assumptions such that those scores from [0,1] can be interpreted as probabilities. Example: weighted k-NN can produce something between 0 and 1, but is not a probabilistic classifier.
Check the definition of probabilistic classifier. As far as I see it, we define it to have scores in [0,1]. But IMO this is not enough and a true prob classifier should have probabilistic assumptions such that those scores from [0,1] can be interpreted as probabilities. Example: weighted k-NN can produce something between 0 and 1, but is not a probabilistic classifier.