By "mind" Anaxagoras meant a cosmic element, not merely the human mind. This can be seen in Socrates' account of his expectations from Anaxagoras, a bit after the passage that you quoted:
And I rejoiced to think that I had found in Anaxagoras a teacher of the causes of existence such as I desired, and I imagined that he would tell me first whether the earth is flat or round; and whichever was true, he would proceed to explain the cause and the necessity of this being so . . . and if he said that the earth was in the centre, he would further explain that this position was the best, and I should be satisfied with the explanation given, and not want any other sort of cause. And I thought that I would then go on and ask him about the sun and moon and stars, and that he would explain to me their comparative swiftness, and their returnings and various states, active and passive, and how all of them were for the best.
That is, Anaxagoras' theory of "mind" related to such matters as whether the earth is flat or round, whether the earth was in the centre of the universe, and as to the movement (and lack thereof) of the sun, the moon and the stars.
Therefore it is plain that Anaxagoras' "mind" was not merely the human one.
As Socrates related further, he was disappointed with Anaxagoras' theory. The mind was supposed to be the source of order in the universe. But it turned out that Anaxagoras' cosmic "mind" was quasi- mechanical, and not sovereign, in the sense that it did not order things according to what it considered to be the best manner.
Socrates did not know, of course, the neoplatonists, but he would apparently be more content with their version of cosmic mind (nous) than with that of Anaxagoras'.
(Edit - some elaboration: Anaxagoras' nous is performing tasks, like rotating and separating the other cosmic elements. Socrates, as reported in the Phaedo, did not consider those tasks directed at what was overall best. The neo-Platonists, on the other hand, followed Plato in depicting the cosmic nous as aiming at the best. Plato placed the Form of the good above all the other eternal Forms, signalling that the Forms represented what was overall best. The neoplatonists placed the cosmic Nous under the cosmic One, which they identified with the platonic Good.)