Sadly, it’s those sorts of statements, which show that Peale took a ‘high’ view of man, that result in the spewing of so much venom against him and his ideas from Christian evangelicals. Actually, Peale was a conservative evangelical for the most part, but he tended not to use, that is, write and speak in, the language of the conservative evangelical.
There is much, much more that the conservative Christians dislike about Peale, including the fact that he was a Freemason, and the fact that he borrowed some ideas, thought-forms and language from New Thought in his sincere attempt to explain, and help people apply, the principles of Christianity. They also attack him for what they see as a more general religious syncretism in his writings and sermons. They dismiss his method of affirmative prayer as nothing other than auto-suggestion. They attack him for preaching both a 'theology of man' and a 'gospel of success,' for his public acceptance and non-condemnation of homosexuality, for appearing to question the truth of certain fundamental (in their view) Christian teachings, for combining religion and pop psychology, and for his having a 'high regard' (Peale's words) for psychiatry. They attack him for allegedly preaching universalism. They somewhat hypocritically attack him for his conservative politics and stances he took on some contentious issues, even though their own politics are almost invariably ultra-conservative. They accuse him of being a bigot---something he definitely wasn't (in that regard, please see this link)---even though they themselves are pretty good at being bigots. They go so far as to attack others who were his friends or who endorsed him or his books, including the Southern Baptist ordained minister and evangelist Dr Billy Graham. Guilt by association, that is---and very unfair. There is no end to it. Even Dr Peale's Wikipedia profile has been sabotaged by these narrow-minded people with an axe to grind so as to accentuate the negative and eliminate, or at least downplay almost to the point of nothingness, the positive about the man and his many achievements. Damn the lot of them, who do this sort of thing, I say. For the most part, they are being grossly unfair to Peale, and they have misinterpreted (I suspect deliberately in the majority of cases) his ideas and writings.
Believed thought is a different matter. Believed thought is thought you have accepted as true irrespective of whether or not it actually is true. Belief adds 'content' to our thoughts and feelings and thus gives them a certain power they otherwise would not have. Believed thought is a matter you do need to be concerned with, for that sort of thought can have biochemical effects on the cells on your body. If you doubt that, I can only suggest you read Bruce Lipton's book The Biology of Belief.
Now, when a negative thought or emotion enters into your consciousness, don’t deny its existence or seek to override or counteract the thought or feeling with a positive one. Again, that is giving the thought or feeling more attention, recognition, and power than it rightly deserves. Observe and briefly note its existence, but spend no time---not even a nanosecond---evaluating, labelling, judging, or condemning that thought. Simply let the thought or feeling go.
In fairness to Dr Peale---well, someone needs to be fair to him---I should mention that in many of his books he spoke about simply 'dropping' negative, unhealthy thoughts as opposed to actively replacing them with positive thoughts. This 'dropping' of unhealthy thoughts, as part of what Peale would refer to as 'emptying the mind,' is much closer to the mindfulness approach. (I recall Krishnamurti referring to meditation as the 'emptying of the mind,' and writing that the mind is empty 'when thought is not.') Peale also wrote about the need for daily practice of what he referred to as 'creative silence.' This again is something quite similar to mindfulness. He also wrote often about the psychological and spiritual principles of indirectness and non-resistance, and the need to stop struggling and let go. Finally, it should also be kept in mind that Peale tended to use the words 'positive thinking' to mean faith in God and Jesus Christ. He saw those things as synonymous, and made that perfectly clear in, for example, his 1980 book The Positive Power of Jesus Christ---a book which gives clear and unambiguous expression to his underlying conservative evangelical faith.